${\cal N}$ = $1$ Euler Anomaly from RG-dependent metric-Background

We consider ${\cal N}=1$ supersymmetric gauge theories in the conformal window. By applying a suitable matter superfield rescaling and a Weyl-transformation the renormalisation group running (matter and gauge field $Z$-factors) are absorbed into the metric. The latter becomes a function of the $Z$-factors. The Euler flow $\Delta a \equiv a_{\rm UV} - a_{\rm IR} |_{{\cal N}=1}$ is then obtained by free field theory computation with the non-trivial dynamics coming from expanding the Euler invariant in the flow dependent metric. The result is therefore directly obtained in terms of the infrared anomalous dimension confirming an earlier result using the matching of conserved currents.


Introduction
For a theory on a curved space in 4D, with no explicit scale symmetry breaking, the trace of the energy momentum tensor TEMT is parametrised by [1,2] where E 4 , W 2 and R are the topological Euler term, the Weyl tensor squared and the Ricci scalar respectively. The Euler anomaly has a long history in relation to the a-theorem, e.g. [3][4][5] and [6] for a review. The theorem states that the difference of the ultraviolet (UV) and infrared (IR) Euler anomaly ∆a ≡ a UV − a IR is strictly positive and intuitively measures the loss of degrees of freedom during the flow. For N = 1 supersymmetric gauge theories an exact expression was derived, valid in the conformal window, [7] almost twenty years ago. Ever since it has served as a fruitful laboratory for testing different techniques by rederiving the result. Examples include four-loop perturbation theory [8], the local renormalisation group (RG) [9] and employing superspace techniques assuming a gradient flow equation [10] conforming an older conjecture [11].
Here we derive ∆a| N=1 by absorbing the matter and gauge Z-factors into the background metric. This renders the theory, in the vacuum sector, equivalent to a free field theory in a curved background carrying the information of the dynamicsg ρλ = f (Z, γ * )δ ρλ . The difference of the Euler anomaly is computed using the conformal anomaly matching and dilaton effective action techniques used by Komargodski and Schwimmer (KS) [5,12] to prove the a-theorem but at the same time differs substantially from it.

Framework
In this section we derive the formula for computing ∆a. Let us consider a massless theory with a generic fields φ and coupling g. The IR effective W is given by where the subscript W on the action stands for Wilsonian. Hence S W is on a renormalisation trajectory from the UV to the IR fixed point. A generic Weyl rescaling of the metric g µν → e −2α(x) g µν results in q 2 → q 2 e 2α(x) and since the correlation functions depend on ratios of q 2 /µ 2 the operation is equivalent to µ → e −α(x) µ. The key idea in [5] is to compensate for this change by coupling a dilaton to the RG scale µe τ(x) which transforms as τ(x) → τ(x) + α(x). The action in (2) then assumes the form S W (g(µ), µ, φ) → S (τ) W (g(µe τ ), µe τ , φ) and is formally dilation invariant. The partition function assumes the form and becomes a function of the external dilaton field. The dilaton is the spurion of the RG transformation and its response (effective action) must be sensitive to the difference in UV and IR conformal anomalies. It was shown in [5] that the relevant term is where the Wess-Zumino action is given by and since ∂ ln µ W τ = d 4 x √g Θ τ (e.g. appendix [13]) one arrives at the key formula where | S WZ denotes projection and Θ τ is the TEMT in the dilaton background. Assuming a conformally flat metricg ρλ = e −2s(τ) δ ρλ the latter reads where it was used that the Weyl tensor vanishes in those spaces since it is sensitive to the spin 2 part only and we discard the H -term since being a total derivative it does not contribute S WZ . Hence determining ∆a reduces to findingã andc which of course depend on the content of the theory. Before turning to the N = 1-computation it is instructive and useful to first discuss the case of a single scalar field.

The scalar field with dynamics absorbed in the metric as an example
Consider a scalar field theory in flat space with Wilsonian action

CONF12
where it is going to be sufficient, for later purposes, to focus on the kinetic term only. The dilaton is implemented by Z(µ) → Z(µe τ(x) ) and it is the key idea of this paper that the entire RG running Z(µe τ(x) ) can be absorbed into metric by the Weyl rescaling with α(x) = s(x) given by The theory then becomes a free field theory S (τ) λ φ on a conformally flat space with metricg µν (9). In the previous formula D (s) ρ = ∂ ρ − (∂ ρ s) are covariant derivatives ensuring local Weyl invariance, a technical detail for which we refer the reader to [13].
To get the Euler anomaly we then need to determine (7) in a free field theory and expand the corresponding geometric term which will involve the dynamics in terms of the anomalous dimension (appearing through derivatives of the absorbed Z-factor). The TEMT in a free theory (cf. [1,14] or the explicit computation in appendix [13]) is given by where the absence of the R 2 -term is a consequence of the free field theory being a conformal field theory. As previously explained the R-term does not contribute to S WZ and can therefore be discarded. Evaluating the latter in the background (9) leads to where the following abbreviationγ ≡ d d log µ γ was used as well as Note that 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 in S WZ (5). Furthermore it is then clear that the total derivative terms in (11) can be discarded since they do not contribute to the bulk-term S WZ (5). In order to project on S WZ it is convenient (following [5], [12]) to set τ = (∂τ) 2 under which S WZ → d 4 x 2( τ) 2 . Using (6) one gets the simple result from the last two terms in (11). Above γ IR,UV ≡ γ(g * IR,UV ) are the values of the anomalous dimensions at the respective fixed points. Eq. (13) constitutes an important intermediate result for the derivation of ∆a| N=1 .

N = 1 supersymmetric gauge theory
We aim to compute ∆a for an N = 1 supersymmetric gauge theory with S U(N f ) × S U(N f ) flavour symmetry and gauge group S U(N c ). In the superfield formalism the action takes the form, e.g. [15], with vector V and matter (Φ f ,Φ f ) superfields. Above W 2 is the supersymmetric gauge field kinetic term (function of V), g is the holomorphic coupling constant 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 [16][17][18] which is also of use in deriving the NSVZ β-function e.g. [15] or the appendix of [13]. Using the Konishi anomaly, and a suitable Weyl rescaling the action (14) can be cast into a form that makes it possible to use the previously studied free field theory example.

∆a| N=1 from Dilaton effective Action and Konishi anomaly
Using arguments of holomorphicity it can be argued that the running of the coupling g, of the Wilsonian effective action of the supersymmetric gauge theory (14), is one-loop exact [15,19] and reads is the Z-factor of the 1/g 2 -coupling, γ * the IR squark anomalous dimension and the arbitrary scale µ ′ > µ can be identified with the UV cut-off Λ UV . We emphasise that S W (µ) is µ ′ -independent. The Konishi anomaly 1 allows, by rescaling the matter fields to shift the Z 1/g 2 -factor in front of the matter term where the arbitrary scale µ ′ > µ can be interpreted as a UV cut-off Λ UV . Crucially, the entire RG flow is absorbed into the precoefficientẐ(µ, µ ′ ) in front of the matter term. Eq. (18) is the analogue of the action (8) 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. The external dilaton field is introduced as beforê The crucial point is that the W 2 -term is Weyl invariant where the matter term is Weyl variant which allows to absorb the dynamics into the metric, i.e.Ẑ(µ, µ ′ ). This is achieved by the following Weyl rescaling α = s 2g ρλ =ẽ a ρẽ a λ = e −2s(µe τ (x)) δ ρλ , s(µe τ (x)) = − CONF12 whereẽ a ρ is the vielbein andg ρλ is the background carrying the dynamics and the remaining part is a free field theory. 3 Since the UV scale µ ′ is arbitrary the physical quantity Θ τ is independent of it and we may therefore send it to infinity. Since the geometric termsẼ 4 andH 2 are independent of µ ′ , 4 the form of (7) implies thatã andc are µ ′ -independent and therefore constants. This means thatã andc take on the values at the (free) UV fixed point and the geometric quantities are to be evaluated using the background metricg ρλ . This allows us to largely reuse the computation in section 2.1 as outlined below.
The steps in completing the computation are as follows. Equivalence to the example in the previous section is attained by replacing Z →Ẑ in Eqs. (9) to (13), (following from (21)) implying γ → δγ ≡ γ − γ * and accounting for the correct number of degrees of freedom ν. Note that only matter fields contribute to the latter sinceẐ only stands in front of the matter term. The matter superfield consists of a complex scalar and a Weyl fermion which contribute [1] ν ≡ 2 C-scalar in units of a real scalar field. This number has to be multiplied by by flavour and colour numbers: 2N f (two matter-field per flavour) and N c (the S U(N c ) Casimir of the adjoint representation). Finally ∆a is given by 2N f N c ν∆a| (13) γ UV,IR →δγ UV,IR . With (γ UV , γ IR ) = (0, γ * ) leads to (δγ UV , δγ IR ) = (−γ * , 0) and finally The result (23) is indeed the same as the non-perturbative result quoted in (Eq.4.18) in [7] when taking into account the explicit form of γ * . The formula (23) is valid in the conformal window 3/2N c < N f < 3N c where the theory is asymptotically free in the UV and the acquires a non-trivial fixed point in the IR. Within these boundaries the anomalous dimension γ * takes on the values −1 to 0 and ∆a is manifestly positive in accordance with the a-theorem. Within N = 1 supersymmetric theories the a-theorem has many applications including the so-called a-maximization [21].

Discussion
The main result of this work is the computation of ∆a| N=1 (23) in the new framework where the dynamics, i.e. the Z-factor and γ * the anomalous dimension at the IR fixed point, is absorbed into the metric (21). This renders the computation equivalent to the Weyl anomaly of a free field theory with non-trivial dynamics originating from expanding the curvature term E 4 in terms of the metric (21). A possible advantage of our approach over the original derivation [7] is that the result is directly obtained in terms of the anomalous dimension of the squarks at the IR fixed point rather than identifying the latter indirectly from a result in terms of N C and N f . Let us conclude by saying that the procedure is reminiscent of the AdS/CFT set-up in that dynamics is encoded in the metric.
RZ would like to thank the Confinement'16 organisers and particpants for a memorable event in Thessaloniki.