On the gluon operator effective potential in holographic Yang-Mills theory

The holographic formalism is applied to the calculation of the effective potential for the scalar glueball operator. Three different versions of this operator are defined, and for each we compute the associated effective potential and discuss its properties and scheme ambiguities. Contact is made to earlier attempts to guess this effective potential from the conformal anomaly. We apply our results to the Improved Holographic QCD model calculating the glueball condensate.


Introduction
The AdS/CFT correspondence can potentially address many non-perturbative features of four-dimensional QCD-like gauge theories. As it has become clear in the past 15 years, virtually all the observables one can construct in QCD can be associated to a geometric counterpart in holographic models.
Although string theoretical, top-down constructions only allow today precise first-principle calculations in supersymmetric versions of Yang-Mills/QCD and deformations thereof, a parallel effort has been undertaken in order to construct, using the same holographic dictionary, phenomenological models which may be closer to QCD [1,2,3,4,6], using bulk Einstein gravity coupled to a scalar field and considering full backreacted solutions of the system.
In the same context, it was recently shown in [7] how to write a fully non-linear expression for the generating functional of scalar external sources up to two spacetime derivatives and, by a Legendre transformation, for the quantum one-particle irreducible effective action for the associated classical operator.
In this work we provide an application of the general results of [7] in the context of QCD. We concentrate on one particular observable, which is present in any nonabelian gauge theory: the gauge-invariant, scalar, dimension four, gluonic operator: (1.1) The vacuum expectation value G has been widely discussed in the past. It is one of the main ingredients of the SVZ sum rules [8], which relate expectation values of QCD composite operators to hadronic observables. Various numerical studies have been performed to compute this quantity using lattice gauge theory [9] 1 . In pure Yang-Mills theory in the continuum limit, the expectation value diverges as a −4 , where a is the lattice spacing, and it is also subject to an additive renormalization. Other works concentrated on the temperature dependence of G in the deconfined phase which, contrary to the vacuum expectation value, is unambiguous, and once the vacuum contribution is subtracted, it is related to the interaction measure (i.e. the trace of the thermal stress tensor) [10].
The quartic divergence of the one-point function G is also present in the holographic theory and it arises close to the UV boundary of the asymptotically AdS space-time. Renormalization of this quantity proceeds by the same counterterm that renormalizes the (also quartically divergent) vacuum energy. However, the finite part which is left over is scheme-dependent, and subject to an additive renormalization. Therefore it is not very interesting to compare the (finite) result G (ren) obtained after renormalization to the numerical values one can find in the literature (from lattice computations, or from phenomenology coupled to sum rules), unless one can relate it to other physical quantities or establish a precise map between renormalization schemes.
The AdS/CFT duality however allows the possibility to go beyond the calculation of the (scheme-dependent) value of the condensate, and to compute the full quantum effective potential for a QFT operator O. This is the homogeneous part of the one-particle-irreducible generating functional, (1.2) where J is the source coupling to the operator O and S is the renormalized generating functional for connected correlators, computed by the renormalized gravity on-shell action 2 . The zero-derivative part of functional Γ(O), i.e. the quantum effective potential for the field O, is uniquely determined once the vacuum energy is renormalized to a definite value. Further, the computation can be extended to include higher derivative terms.
In this paper we present the calculation of the quantum effective action for the gluonic operator (1.1) and for its RG-invariant relative, in phenomenological holographic duals of pure Yang-Mills theory constructed from Einstein-Dilaton fivedimensional models, and in particular in the Improved Holographic QCD model [6]. These models are characterized by the presence of a bulk scalar field, which is dual to the source associated to this operator i.e. the Yang-Mills coupling λ. The dual scalar field has a non-trivial dependence on the radial coordinate, which encodes the RG-scale dependence of the coupling. 3 We will compute the effective action up to two space-time derivatives, i.e. in the general form: and we will compute the effective potential V(O) and the kinetic function G(O). We will also present the results for the corresponding canonically normalized field, obtained by integrating the relation dO c = G(O) dO.
In the holographic theory, we show that one can define a non-perturbative, RGinvariant scale Λ [6], which is the geometric analog of the QCD IR scale, and sets the scale of all non-perturbative observables (condensates, glueball masses, etc). We compute the effective potentials both in the bare theory with a UV cut-off Λ U V , and in the renormalized theory at a scale µ. The effective potentials depend on the scale Λ and, depending whether we are using the renormalized or the bare regularized theory, on the RG scale µ or cut-off scale Λ U V , respectively.
In the renormalized theory, we consider the effective potentials corresponding to different but related operators: 1. The first one is the gluonic operator defined in equation (1.1). The source for this operator in the field theory is the 't Hooft coupling, which is scale-dependent. Thus, the renormalized effective potential will depend on the renormalization scale µ, and has the general form: where v is a dimensionless function of its arguments, whose precise form depends on the bulk dilaton potential 4 , and µ is the renormalization scale. The value of the non-perturbative scale Λ is fixed by choosing a particular solution of the RG-flow equations, which in the bulk amounts to fixing all integration constants of Einstein's equations.
Beside being scale-dependent, the effective potential for T rF 2 depends crucially on the relation between the field theory 't Hooft coupling λ and the bulk scalar field. This is one of the sources of ambiguity of phenomenological models, as it is not fixed from general principles. In the UV, this ambiguity can be fixed to a certain extent, by matching the holographic model to Yang-Mills perturbation theory. However, changing how we identify the coupling in the IR can be interpreted as a source of scheme dependence, and it modifies the shape of the potential. The precise identification of the coupling can only be fixed if the gravity dual is embedded in a top-down model.
Thus, although the calculation of Γ(G) can be performed, it is a nontrivial issue to relate the result to field theory non-perturbative observables, without a clear identification of the coupling with a bulk field at all scales. This drawback is avoided if we consider the renormalization-group invariant gluonic operator, as we discuss next.
2. The renormalization group invariant version of the gluonic operator G is defined by: where β(λ) is the beta-function for the running 't Hooft's coupling λ. This operator is related by the conformal anomaly to the trace of the stress tensor, T µ µ = T . Unlike the case described in point 1 above, the form of the effective potential for T is universal, it does not depend on the identification of the coupling, nor on the precise form of the bulk dilaton potential, and it can be written in closed analytic form: where Λ is the holographic non-perturbative scale. Due to the RG-invariance of the operator T , the potential is itself independent of scale. Its expression is universal up to the scheme-dependent vacuum energy renormalization, which in writing equation (1.6) was chosen in such a way that V(T ) is extremized exactly at the non-perturbative scale Λ: Equation (1.6) reproduces the field theory trace anomaly, and in a sense it is an exact, model-indepednent prediction of holography applied to QCD.
In the UV, at small coupling β(λ) ∼ −b 0 λ 2 , and the operators (1.1) and (1.5) coincide up to a multiplicative constant. In fact, a UV expansion of our computation shows that, as µ Λ, the effective potential (1.4) reduces to the result (1.6).
3. Going beyond the effective potential, we consider the two-derivative term in (1.3), which can be obtained from the general results in [7], and derive effective actions for the canonically normalized operators corresponding to G and T . The corresponding effective action up to two derivatives (and omitting the Einstein-Hilbert term) has the form: where O (c) stands for either G (c) or T (c) . In both cases, the form of the potential V in this case is not universal, but depends on the details of the bulk theory.
For the operators discussed above, we also compute the bare effective action, calculated in the theory regularized at a cut-off scale Λ U V . This is a useful quantity to compare with lattice calculations before renormalization: it is independent of the renormalization procedure, which may be difficult to translate between holography and other techniques. The quartically divergent effective potential in the bare theory takes the general form: where O stands for either G or T . In the equation above, Λ U V sets the cutoff scale, whereas Λ is set by the choice of the initial condition (the value of the source/coupling) at the cutoff. The cut-off is a priori arbitrary, but for the theory to be perturbative in the UV (as in standard QCD) one must require that Λ/Λ U V 1. In order to make more explicit the connection with field theoretical computations, we also provide the explicit form of the subtraction of the vacuum energy, as a function of Λ U V and Λ, necessary to go from the regularized to the renormalized theory.
Except for the exact RG-invariant effective action (1.6), the full effective potentials we discuss can only be found numerically, once a concrete bulk theory is specified. In the final part of the paper we provide explicit numerical results for the effective potentials (1.4) and (1.9), and compare them with the universal function (1.6), in a concrete theory: the Improved Holographic QCD model (IHQCD), reviewed in [6], which was shown to lead to realistic, quantitative agreement with lattice Yang-Mills spectra and thermodynamics, [4].
There are several aspects of our computation that are of interest, both for QCD application and more generally for the phenomenology of strongly interacting field theories. On the QCD side, although we are not aware of a lattice calculation of the effective potential of T rF 2 , such a calculation should in principle be possible, and it would be interesting to compare the result to the one found here after the ambiguiity in the normalization of the extremum of the potential are removed.
It has been proposed that a strongly coupled sector may account for the dynamics observed in nature (for example for electroweak symmetry breaking, or for inflation [11]), and holography has been often used as a model-building tool to describe the strongly coupled sector (see e.g. [12] for a recent investigation of holography in the context of technicolor-like theories). Our results for the effective action for the canonically normalized operator, including the kinetic term, can serve as the starting point for phenomenological studies in theories where the vacuum dynamics is not driven by an elementary field, nor by (composite) particle-like excitations, but rather by the dynamics of a condensate. In fact, the effective action (1.8) should not be interpreted as describing particles arising from the quantization of O, and it is not to be confused with e.g. the effective actions of particle states (glueball excitations) in a strongly coupled theory. It is therefore quite different from those described in e.g. [13], where the attention was focused on particle-like excitations in holographic theories, and the computation of properties such as their mass.
The structure of this work is as follows. In Section 2 we review the holographic five-dimensional Einstein-dilaton setup used to describe Yang-Mills theory, we provide the holographic dictionary, and we give the holographic definition of the nonperturbative scale Λ.
In Section 3 we define the various operators we consider and provide the general procedure to construct the (renormalized or regularized) quantum effective actions. We also give the explicit form of the subtraction procedure used to renormalize the vacuum energy.
In Section 4 we give the results for the effective potentials, computed numerically in the concrete IHQCD background.
In Section 5 we offer some concluding remarks and perspectives. Technical details of the UV and IR form of the effective potential for T rF 2 are left for the Appendix.

Notation and conventions
Throughout the paper we will work in the Euclidean signature, in which the appro-priate definition of the Legendre transofrm is given by (1.10) and the relation between sources and one-point functions is: δO (1.11) (no extra i's needed, as it is the case instead in the Minkowski signature). Given the Euclidean effective action in the form (1.3), the corresponding Minkowski effective action is readily written as: with η µν = diag(−1, +1, +1, +1) and the same functions V(O) and G(O) as in (1.3). We denote boundary space-time indices by µ, ν . . ., bulk space-time indices by a, b, . . ., we use g ab for the bulk metric and we keep the notation η µν for the flat euclidean metric.

Yang-Mills theory and Einstein-dilaton gravity 2.1 The Model
In holography, pure 4d Yang-Mills theory at large N c is expected to be dual to a 5d string theory (or a classical gravitational theory in the low energy limit). There is one extra dimension in the dual theory because of the existence of a single adjoint vector in the boundary YM theory [5,6].
The gauge invariant single-trace operators and the bulk on-shell states are in one-to-one correspondence. The most relevant degrees of freedom are the lowest dimension gauge-invariant operators T r[F µν F ρσ ], which can be decomposed according to spin into: the stress tensor T µν ∼ T r[F µρ F ρ ν − 1 d η µν F 2 ], the YM operator T r[F 2 ] and the topological density T r[F ∧F ]. In the holographic dual, the related bulk fields are the 5d metric g µν (dual to the stress tensor T µν ), a scalar field φ (dual to the YM operator T r[F 2 ]) and an axion field dual to T r[F ∧ F ]. In the following, we will ignore the axion because the contribution to the QCD vacuum from the topological sector is suppressed by 1 Nc in the large N c limit. To be specific, we consider a bulk theory with the following five-dimensional action: where S bulk are the bulk terms: and S GH is the boundary Gibbons-Hawking (GH) term: In the above expressions, the coordinate u is the radial (or holographic) coordinate, and {x µ } µ=0...3 are the space-time coordinates on constant-u hypersurfaces. M is the five-dimensional Planck scale. The number of colors N c sets the magnitude of the five-dimensional Planck scale: in the large N c limit we assume that while the scalar potential V (φ) stays of order one, so that classical gravity can be used as a reliable approximation. The bulk scalar potential encodes all the information about the RG flow. The holographic direction, which is related to the energy scale of boundary field theory, is parametrized by the u-coordinate. The ultraviolet and the infrared endpoints of this coordinate are denoted by u U V and u IR , which may be the physical U V and IR fixed points of the full theory, or the U V and IR cutoffs. In the Gibbons-Hawking term, γ µν is the induced metric on the slices and K = γ µν K µν is the trace of the extrinsic curvature. Here and in the following discussions, the subscripts U V or IR indicate that the quantities are evaluated on the U V or IR slices.
Vacuum solutions which preserve Poincaré invariance have the general domainwall form (up to diffeomorphisms): Solutions of this form may be found by first specifying a superpotential W (φ), i.e. a particular solution of the equation: where = d dφ is the derivative with respect to the scalar field. Once W is chosen 5 , the functions (A(u), φ(u)) are solutions of the first order system: The solution (together with a specified initial condition 6 ) may be put in the form: The superpotential should be chosen in such a way as to satisfy appropriate IR regularity conditions. For example it should be such that the solution allows small black hole deformations [14], and/or that the fluctuation problem is well-posed without the need of extra IR boundary conditions [15]. For monotonic potentials these criteria specify the solution uniquely [3]. 6 Due to the shift symmetry in u of the system (2.7) one can impose this initial condition at an arbitrary u without physically changing the solution, i.e. different choices of the initial point can be related by a diffeomorphism. What is invariant is the value of A at a given φ.
We will consider solutions that have a UV-asymptotically AdS region, where This equation sets the asymptotic AdS scale, which appears in the asymptotic value of the potential (and of the superpotential): (2.10)

The holographic dictionary and the potential
Given a solution of the form (2.5) , the holographic dictionary between gravity and field theory quantities is the following: 1. The field theory energy scale µ is identified with the metric scale factor e A , up to a multiplicative constant, which we can choose to be the asymptotic AdS scale: This is the appropriate (and reparametrization-invariant) translation between the radial coordinate and the energy scale [7].
2. We define the field λ(r) as: As discussed in [6], this field is identified with the running 't Hooft coupling of the QFT 7 . As λ → 0, i.e. in the UV limit, the identification with 't Hooft coupling is accurate. However, as discussed in detail in [4], this assumption may not hold in the IR, and an extra field redefinition may be necessary to relate the bulk field φ(r) to the gauge coupling. This introduces an ambiguity in the calculation of IR quantities, like the quantum effective potential for T rF 2 , which relies on a particular identification of the 't hooft coupling with a particular bulk quantity.
However, as we will see in section 3.3.2 the effective action for an appropriate RG-invariant gluon condensate will be independent of the identification of 't Hooft coupling with a specific function of the bulk field φ.
3. The holographic β-function is obtained from (2.7) and (2.11-2.12): In [6] the scalar field is not canonically normalized and the 't Hooft coupling is λ = e φ . Here the scalar field φ is canonically normalized, which accounts for the additional factor Using equations (2.8), (2.11) and (2.13) we recover the usual integrated RG-flow relation between the scale and the running coupling, . (2.14) In the following, we will review the choice of the bulk scalar potential for realistic holographic models like IHQCD [6].
Let us start with the small-λ behavior of the potential. In the UV, an asymptotically free theory is also asymptotically scale invariant. Thus the corresponding bulk geometry should tend towards AdS 5 spacetime, and the bulk potential should asymptote to the cosmological constant 12/ 2 . Then the coefficients of the UV expansion around the constant are fixed by the perturbative β-function through equations (2.13) and (2.6).
Concretely, in Yang-Mills theory the 1-loop β-function is where b 0 = 11 24π 2 . The first two leading terms in the UV expansion of the superpotential W (λ) are therefore fixed by matching the holographic β-function (2.13) and YM β-function (2.15): The UV-asymptotic form of the bulk scalar potential is determined by the above equation via equation (2.6): At large λ, we consider potentials with the rather general asymptotic behavior: In the IR, the bulk potential should give rise to confining geometry characterized by the area law of the Wilson loop, which indicates a linear potential between two quarks. Requiring confinement, a mass gap and a discrete Regge-like glueball spectrum fixes the asymptotic form of the bulk potential in the IR to be (2.18) with [1]: The corresponding IR-regular solution of the superpotential is: where

The non-perturbative scale
In four-dimensional Yang-Mills, the fundamental parameter that defines the theory is not the coupling constant, which depends on the energy scale, but rather the value of the non-perturbative RG-invariant scale Λ which sets the scale of all dimensionful observables. A choice of the value of Λ is in one-to-one correspondence with an RG flow trajectory, i.e. a particular solution of the β-function equation, A choice of Λ is equivalent to a choice of initial condition λ(µ 0 ) = λ 0 for equation (2.22).
In perturbation theory, we can define Λ using e.g. the one-loop β-function, for example, as the scale at which the coupling becomes infinite. Explicitly, using β 1−loop (λ) = −b 0 λ 2 , and integrating equation (2.22), we obtain the usual definition: which is constant along an RG-trajectory governed by the one-loop beta-function, with initial conditions specified by λ(µ 0 ) = λ 0 . As we anticipated, the value of Λ is in one-to-one correspondence with a choice of the coupling at a given scale µ 0 . Of course, the definition (2.23) is not exact, and higher order terms in the β-functions will introduce a µ-dependence in the right-hand side. To define a truly RG-invariant scale, we notice that the quantity is RG-invariant for any choice of a reference value λ * . Changing this reference value only changes Λ by a multiplicative constant, which we can fix by relating the expression (2.24) to the value of a physical observable, for example an RG-invariant operator. This will be done in the following sections 8 . Again, specifying a value of Λ in (2.24) picks a special solution of the RG-equation (2.22), determined by the initial condition λ(Λ) = λ * , i.e. it picks a unique physical theory.
On the gravity dual we have exactly the same situation: a choice of the superpotential W (λ) fixes the holographic β-function via equation (2.13), and an RGtrajectory is fixed by further specifying an initial condition for equation (2.7). Given such a solution, we can define a quantity with the dimension of a mass scale, which is constant along the radial flow: first define the scalar function: This quantity is a function of φ and of a reference point φ * , whose specific value affects A only by an additive constant. By equation (2.8), when evaluated on a solution φ(u), A(φ(u)) coincides with the metric scale factor, up to an additive constant. Therefore, the quantity: where ξ is an arbitrary dimensionless parameter, is u-independent along any domainwall solution like (2.8).
In terms of λ = exp[ 3/8φ], the function defined in (3.3) becomes: where in the second equality we have used the expression of the holographic βfunction in terms of the superportential, equation (2.13). Recall also that e A(u) is identified in the holographic dictionary with the energy scale µ. Thus, equation (2.26) reproduces, up to a normalization constant, the field theory expression (2.24). Finally, if we take the perturbative approximation for W (λ) in (2.16) and we stop at O(λ 2 ), we recover the standard one-loop expression (2.23).
In order to completely specify the holographic version of the field theory scale Λ Y M , we have to choose the constant ξ and the reference point φ * (these two parameters are in fact redundant, since we can always reabsorb a shift in φ * in a redefinition of ξ). For later convenience, we fix this ambiguity by choosing: Since, as we discussed earlier, M 3 ∼ N 2 c , the above choice ensures that Λ stays finite in the N c → ∞ limit. As we will see, with this choice Λ matches the scale of the glueball condensate.
With the definition (2.28), the non-perturbative scale is specified by: • Two parameters entering in the bulk action, namely M and ≡ lim φ→−∞ 12/V (φ).
• The parameter A * = A(φ = 0) specifying the initial condition for the bulk solution.
The product M can be fixed for example from the high-temperature regime of the theory [3]: We finally arrive at the definition: This definition still has a one-fold degeneracy in choosing the holographic model dual to a Yang-Mills theory with a given scale Λ. However, notice that A * and are not separately observable, since ultimately affects only a choice of energy units. Therefore, for a given physical scale Λ, we are free to choose to be any reference scale, and fix the integration constant A * to reproduce the physical value of Λ. For a thorough discussion about this point, see [4].

Holographic effective actions
In a previous work [7], a general formalism was presented where the holographic RG flow and the quantum effective action are derived from the gravitational dual.
In holography, the generating functional of the connected correlators is equal to the action of the bulk theory evaluated on-shell. This is divergent due to the infinite volume of the bulk, and these divergences parallel the UV infinities of the dual quantum field theory As in field theory, one can define a finite, regularized generating functional in a cut-off version of the theory, by imposing a (large-volume) cutoff on the radial coordinate u.
As it was shown in [16,7], once a suitable IR regularity condition is imposed on the superpotential, the regularized on-shell action is a UV boundary term which depends on the scalar field and induced metric at the cutoff, and has the form: is the superpotential, and the function U (φ) that enters the second-derivative terms is given by: with the function A(φ) defined as: where φ * is a reference value of the scalar field and its choice amounts to fixing the integration constant in A(φ), but it does not change equation (3.2). By equation (2.8), when evaluated on the solution φ(u), the function A(φ(u)) conicides up to a constant with the scale factor A(u) (in particular, it has the same leading UV asymptotic behavior). As discussed in [7], a way to translate the holographic cut-off into the dual field theory language is to notice that, in the homogeneous case, the scale factor e A(u) of the induced metric corresponds to the field theory energy scale, and the UV corresponds to the limit e A → ∞. If we cut-off the radial coordinate at a point u = u U V , we can define the energy cut-off Λ U V by: where in the last line we have written the cutoff in terms of the small parameter typically used as cut-off for the conformal coordinate r of Poincaré AdS, where e A ≈ /r in the UV.
In the UV, as A → ∞, or λ ≡ e √ 3/8φ → 0, the functions W and U go to constant values, On the other hand, writing the induced metric in the UV as in terms of boundary metric γ (0) µν which stays finite in the UV limit, the regularized on-shell action (3.1) becomes: where the leading dependence of the cut-off is manifest, and we have written the action as a function of the 't Hooft coupling λ at the cut-off.
To obtain finite results once the UV cut-off is removed, one needs to perform the renormalization procedure. As in a renormalizable theory, one identifies all the UV divergent terms and subtracts them by adding a finite number of counterterms on the boundary.
In [7] an explicit expression was found for the renormalized generating functional depending on the induced d-dimensional metric and the scalar field, up to two derivatives: where φ and γ µν are evaluated on an arbitrary slice in the bulk, and the coefficient functions are: The constants D 0 , D 1 , D 2 , which are dimensionless, are determined by the difference between the UV-finite terms in (3.7) and the corresponding terms in the counterterm action, which takes the same form as (3.7) but with different functions W (φ) and U (φ) [7]. The constants D 0 , D 1 , D 2 completely parametrize the schemedependence which arises from choice of counterterms with up to two derivatives. We will discuss this point further in Section 3.

The function
Finally, the functions G 0 (φ) appearing in (3.10-3.11) are defined as: where the integration constants G 0 (φ * ) and G (2) 0 (φ * ) are chosen in such a way that: From (3.1) and (3.8), one can derive bare vacuum expecation values in the cut-off theory, or the renormalized vacuum expectation values, respectively. In general, we will deal with operators whose source is a function J(φ), rather than φ itself. The corresponding (bare and renormalized) expectaction values are: , .
(3.15) The first will depend on the coupling at the cutoff, and also explicitly on the cutoff scale; the second is finite as the cut-off is removed, and will depend on the choice of the coupling at the fixed renormalization scale µ set by the scale factor in γ µν .
The 1PI action, or quantum effective action Γ( O ) is obtained by Legendre transforming the renormalized generating functional (3.8) with respect to the source J. We can compute the effective action in the regularized theory starting from (3.1), or the renormalized effective action starting from (3.8). In either case, we use the definition 9 : where J is written as a function of O by inverting equation (3.15). The physical vacuum expectation value for vanishing source is found, as usual, by extremizing the effective action: In the following sections, we will separately discuss the bare and renormalized effective actions for two different operators, i.e. the Yang-Mills operator T rF 2 and its renormalization-group invariant version, −β(λ)/2λ 2 T rF 2 , which coincides with the trace of the stress tensor via the conformal anomaly. Before we enter the detailed discussion, we make a few remarks about the general structure of effective actions obtained from the above procedure.
To lowest order in derivatives the vev of the renormalized dual operator is obtained by the variation of the zero-derivative term in the generating functional (3.8) with respect to the source J: where Z 0 is in (3.9). This receives derivative corrections from the derivative terms in (3.8). However, as shown in [7], the general expression for the renormalized effective action up to two derivative order is rather simple: where J 0 (O) is the inverse function of (3.15) determined at zeroth-order only, i.e. it coincides with the zero-derivative term of the full source J(O) as a function of the vev O . In other words, the derivative corrections to (3.15) are cancelled by similar twoderivative terms in S (ren) , and (3.19) is the complete result to two-derivative order. In Section 3.3, we apply this to the compution of the effective action of gluonic operators G and T given respectively by (1.1) and (1.5).

The subtraction scheme
Before extracting explicit results for the glueball operator effective potential in QCDlike models, in this section we will give a few details about the subtraction procedure we use to obtain the renormalized effective action, and how it may relate to schemes adopted in standard quantum field theory. General renormalisation of Einstein-Dilaton theories was developed in [16]. Here, we will try to answer as explicitly as possible the question, what is the subtracted part of the generating functional in the holographic scheme, in terms of physical quantities that one can relate to the field theory?.
We will concentrate on the zero-derivative term in the action, whose subtraction corresponds to an additive renormalization of the vacuum energy. Similar considerations apply to the two-derivative terms, which correspond to a renormalization of the Einstein-Hilbert term and of the kinetic term associated to a space-dependent coupling.
At the zero-derivative order, the regularized generating functional is the vacuum energy in flat spacetime, calculated in the cut-off theory and it is given by the first term in equation (3.7): The energy cut-off Λ U V in the holographic scheme is related to by Notice that, in the definitions above, we could have avoided any reference to the UV value of the conformal coordinate u U V : the coordinate-invariant data are, in a given solution, the value λ of the dilaton when the scale factor takes on a given value 1/ . Similarly, since in the coordinates (2.5), the warping factor A enjoys a shift symmetry, the only invariant way to decide whether the cut-off is actually in the UV is whether the value of the coupling at the cut-off is small since, in the UV-complete solution, λ → 0 as e A → ∞. Thus, at the cut-off we choose This is similar to the situation in ordinary QCD, where the theory is taken to be perturbative, and the coupling small, at the UV cut-off scale. As we will see below, as in QCD, this implies a parametric hierarchy between the UV cut-off and the non-perturbative scale Λ defined in Section 2.3, Λ/Λ U V 1. One can change the cutoff by changing and λ at the same time, by following the flow of the chosen solution (A(u), λ(u)). The limit → 0 corresponds to removing the cut-off, and can be taken only after a subtraction is performed: in fact, W (λ) has a finite value at λ = 0, thus by equation (2.16) the regularized vacuum energy (3.20) diverges as To obtain the renormalized on-shell action one has to add boundary counterterms at the UV slice. At the zero-derivative order, the appropriate covariant counterterm is [17,16,7]: where W ct (λ) is an arbitrary solution of the superpotetial equation (2.6) which flows to the same UV-AdS fixed point, and in the last equality we have used Choosing W ct = W will subtract the divergence but leave a non-zero finite result. In fact, as discussed for example in [3,18,7], for small λ any solution of equation (2.6) takes the form: where C is an arbitrary real number and W 0 (λ) and W 1 (λ) are universal functions, and the order of the subleading terms will be specified later. In particular, if the bulk potential V (λ) has an analytic expansion around λ = 0, then W 0 has an analytic expansion in integer powers of λ, with all coefficients w n determined by the expansion coefficients of V (λ) around λ = 0 [3] (the first two terms are given explicitly in equation (2.16). The function W 1 is determined by W 0 and is given by: (3.28) The above equation defines W 1 up to an integration constant, but given that C in (3.26) is arbitrary we can choose this constant at will. Using the power-law expansion for W 0 (λ) in equation (3.28) it is easy to obtain explicitly the small-λ expansion for W 1 : this consists in a power series similar to (3.27), multiplied by an overall non-analytic factor: are the first two beta-function coefficients [3], and the power series coefficientsw i are determined by w i except for a common overall factor. One can go further in (3.26) by adding the subleading order terms: these are proportional to C 2 and to the square of the non-analytic exponential in λ [18]. It is instructive to compare equation (3.28) with equations (2.26-2.27): we see that, up to the choice of a multiplicative constant which we can choose by rescaling C to be related to ξ defined in (2.28), we have: Thus, at the cut-off surface, we can write both W (λ ) in (3.20) and W ct (λ ) in (3.25) in the same form (3.26), the only difference being encoded in the contants C, C ct multiplying W 1 : where the O( 8 ) terms are corrections to the leading terms in (3.26). From (3.20) and (3.25) we see that the renormalized action is: The coefficient D 0 appearing in (3.9) is then given by Thus, holographic renormalization of the vacuum energy proceeds via an additive renormalization by a function of the cut-off (including divergent and finite terms): In the above expression, the dependence on the cut-off is both in the explicit 1/ 4 and in the dependence through λ . We can take one step further and express the subtracted function of the cutoff purely in terms of physical quantities. Although a close analytic expression cannot be obtained, this can be done explicitly order by order in a log-expansion, as we show below.
In the UV, we can express λ as a function of the cut-off and the nonperturbative scale Λ, by using equations (2.26) and (2.27): for small λ, Evaluating this expression at the cut-off u U V , using equation (2.26), recalling that A(u U V ) = log( / ) and neglecting subleading terms 10 , we have the approximate relation: where ξ is defined in equation (2.28) and we have replaced by Λ U V using (3.22). Thus, using equation (3.37) we can have an expression, written as a power series, for the function of the cut-off that we are using in the subtraction, in terms of the phyisical parameters, i.e. the cut-off scale Λ U V and the RG-invariant non-perturbative scale Λ: (3.38) In this expression, the divergent terms are uniquely determined by the expansion coefficients w n of the leading superpotential (3.27), which are in turn uniquely determined by the expansion coefficients of the bulk potential V (λ) around λ = 0. The same can be said for the universal subleading terms, which correct each term in the series at O(log log(Λ U V /Λ) / log(Λ U V /Λ)). The finite, non-universal term depends only on an overall coefficient C ct and determines the renormalized vacuum energy in terms of the non-perturbative scale.
Equations (3.32) and (3.38) make the subtraction scheme manifest in terms of 1) the physical quantities, i.e. the cutoff scale Λ U V and the non-perturbative scale Λ associated to the solution; 2) a scheme-dependent constant C ct , which determines completely the choice of the counterterm.
As a final remark, notice that, by equation (3.37), requiring λ to be small at the cutoff is the same as requiring that which is the usual condition on the separation of the UV scale from the IR scale for QCD to be perturbative at the cutoff.

Effective actions for glueball operators
There are several different composite operators that one can associate to the Yang-Mills field strength. They have different effective actions, which arise by Legendretransforming the generating functional with respect to different source functions J(φ). We will consider the following choices.
1. One of them is simply G ≡ T rF 2 . As we discuss below, this is obtained by choosing the source J = −1/2λ. Since the generating functional is RGinvariant, but the coupling is scale-dependent, this operator is itself non-RGinvariant. Moreover, in the holographic theory it depends on the relation between the field theory 't Hooft coupling and the bulk field λ, which may be non-trivial in the IR.

The RG-invariant version of the gluon composite operator is
This coincides with the trace of the stress-tensor, and as we will see in Section 3.3.2 its vev can be obtained by taking the source to be proportional to the function A(φ) in (3.3), i.e. the scale factor. This operator is universal, and its potential does not depend on the identification of the t'Hooft coupling in the bulk.
3. The VEVs and effective actions of the operators above can be obtained in both the bare regularized theory and the renormalized theory, depending whether one takes (3.7) or (3.8) as a starting point.

4.
Including second order derivative terms in the effective action, we may also consider redefinitions of the above operators such that they are canonically normalized. In the case of renormalized operators, by equation (3.19) we see that the kinetic term has the universal form: where φ 0 (O) is the zeroth-order relation between φ and O obtained from (3.15) and Z 2 (φ) is given in equation (3.11). Thus, the canonically normalized oper-atorÕ is defined by the relation The same consideration applies to the bare, regularized operator, in which case one has to substitute Z 2 (φ) with the function W U /W , see (3.7).

The renormallized composite gluon operator T rF 2
The Lagrangian of Yang-Mills theory is where λ = g 2 Y M N c is the 't Hooft coupling. In the following, we will absorb the factor N c into the normalization of the vector potential and define the operator coupled to the 't Hooft coupling as the YM operator Thus, expectation values of (products of) G can be obtained by taking functional derivatives of the quantum generating functional with respect to J = (−4λ) −1 . We can then make the coupling space-time dependent and use it as a source to perform the Legendre transform: The minimum of this effective potential is in the far infrared where both the vev G = T rF 2 and 't Hooft coupling λ go to infinity: which sets the source J min ∼ λ −1 min to zero. Therefore, after a deformation around the UV fixed point, the vev will flow to the minimum of the effective potential in the strongly coupled IR. In this standard effective potential, one can only deduce the properties of the IR physics, not the information in the intermediate scale between the UV and IR.
To study the RG scale dependence of the effective potential, it is more appropriate to use a modified version of the Legendre transformation in which the source, instead of being the full coupling, is taken to be the fluctuations of the coupling around a background value.
In a QCD-like model, at any finite intermediate scale µ, the background 't Hooft couplingλ(µ) is finite, thus the background sourceJ is non-vanishing. In this modified computation, we will perform the Legendre transformation with respect to the fluctuationJ(x) = J(x) −J(µ), namely the difference between the full coupling J(x) and the background couplingJ(µ) at the RG scale µ. The effective action is given by whereJ(x) = J(x) −J(µ) andJ = (−2λ) −1 . Using this definition, the minimum of the effective potential (the zeroth order term in the effective action Γ[G, µ] ) will correspond to a finite vev at the RG scale µ corresponding to a finite couplingλ(µ): .

(3.48)
Due to the scale dependence of the background value, both the quantum effective action and the expectation values will depend on the renormaliztion scale µ. Now let us compute the full effective action from the Legendre transformation (3.47). To zero-derivative order, the YM operator G = T rF 2 , as a function of µ and λ, is computed by the functional derivative of the renormalized generating functional (3.8) with respect to the 't Hooft coupling (3.49) In this calculation, we have identified the energy scale µ with the scale factor of the induced metric as in (2.11). The vev calculated here is the same as the standard definition because the variation ofJ(x) coincides with that of the full coupling J(x).
To derive the full coupling λ(G) as a function of the vev, one first inverts the relation between λ and G λ = λ G µ 4 . We will present here the analytic results for the UV and IR, where one can expand the superpotential for small and large λ respectively. We mean by the UV limit that the RG scale µ is much larger than the other dimension-one quantities while the IR limit indicates a hierarchy between the RG scale µ and the other dimension-one quantities in the opposite way The details are presented in Appendix A.
Using the asymptotic form of the superpotential in the UV and IR, we calculate the effective action in these two limits. The UV effective action is given by where we used the parameter ξ defined in equation (2.28), i.e.
In order to fix all the coeficients we have chosen a renormalization scheme where the constants D i in the renormalized generating functional (3.8), (3.9-3.11) take the following values: This choice of D 0 , as we will see in the next subsection, sets the minimum of the effective potential of the RG-invariant glueball operator at the value Λ. The definition of canonically normalized operator depends on the coefficient of the kinetic term and thus D 2 . The value of D 2 was chosen for convenience to simplify the coefficients in the effective action. In the numerical calculation of section 4, we will use another value of D 2 to set the minimum of the effective potential of the canonically normalized operator at the value Λ. A non-zero value for D 1 would make the Ricci scalar enter the action second order in derivatives, which would then mix with the scalar kinetic term. Since we are mostly interested in a Minkowski background, we have chosen to set its coefficient to zero. Now we can canonically normalize the operator according to the kinetic terms: The effective action of the canonically normalized vev G (c) in the UV is where we have neglected subleading terms in the potential.
In the IR (µ G, µ Λ), the effective action reads where: Results for generic values of Q and P can be found in the Appendix. The canonically normalized operator G (c) is determined by the kinetic term The corresponding effective action for the canonically normalized operator in the IR reads where we have defined: After presenting the analytic results in the UV and IR where perturbative expansions are possible, we are going to compute the full non-perturbative potential by numerically solving the equations. As an illustration, we consider a bulk scalar potential V (λ) = 12 2 1 + 11 27π 2 λ + 1 100 λ 4 3 log(1 + λ) 1 2 , which has the correct UV and IR asymptotic behaviour given by (2.17) and (2.18) with V ∞ = 3 25 . The 1/100 factor in front of the third term is to disentangle the UV behaviour from that of the IR.
We present the result of the numerical calculation of the effective potential of G in figure 1. In the extreme UV region, the potential approaches the analytic form (3.56) in the limit µ → ∞, and it becomes steeper as we lower the RG scale µ; there is a crossover region around Λ/µ ∼ 10 5 where this trend is inverted, and the potential starts flattening again as we lower the RG scale. This flattening continues all the way towards the deep IR where, as we discussed below equation (3.46), the minimum of the effective potential moves off to infinity.  Figure 1: The non-perturbative effective potential of G. In the UV limit µ → ∞, the potential approaches the analytic form (3.56) derived from the UV expansion. For large G in the IR limit µ → 0, the potential has a linear dependence on the vev G in accord with the IR expansion result. As we decrease the RG scale µ, the potential goes up in the UV region, then slows down the trend of going up, and finally goes down in the IR region.

The RG-invariant gluon operator
In the previous section, we have discussed the quantum effective action for T rF 2 . However, notice that T rF 2 is not RG-invariant and there are ambiguities in the identification of the 't Hooft coupling λ coupled to T rF 2 in the gravity dual when we go to the IR. An RG-invariant gluon operator is of particular interest due to its RG scale independence. One of the simplest RG-invariant operator in Yang-Mills theory is the combination appearing in the trace identity, Using the operator on the right hand side one can define an RG-invariant gluon condensate, whose value is proportional to the non-perturbative scale Λ of the theory. We will concentrate on this RG-invariant gluonic operator in the discussion below.
In the holographic theory, the RG-invariant operator T associated to the bulk scalar will be coupled to an appropriate source function J inv (λ(x)), rather than to J = (−2λ) −1 . Then, the vev of T is the functional derivative with respect to J inv of the renormalized generating functional S (ren) [γ µν , λ(J inv )], given in (3.8), thought as a function of J inv . Keeping only the zeroth order term in derivatives, we have where µ = −1 e A is the energy scale (see equation (2.11) ). At the zero-derivative order, the quantity µ e −A coincides up to a normalization constant with the non-perturbative scale (2.26) and is itself RG invariant. Thus, the constancy of the vev T requires dA/dJ inv to be constant. We define: With this definition (3.70), we see that T coincides with the standard RGinvariant gluon condensate operator in Yang-Mills theory : where in the third equalities we used equation (2.13). Notice that T is positive since the β-function is negative. It is important to notice that the relation (3.71) that relates T and − β(λ) 2λ 2 T rF 2 is independent of the identification of the bulk field λ(u) with the Yang-Mills coupling. As discussed in [4], this identification can be established in the UV, but it could change in the IR, which introduces an extra scheme dependence in the holographic setup and makes it difficult to relate it to ordinary Yang-Mills theory. However, this ambiguity does not affect the relation (3.71). Indeed, suppose we relax the indentification of λ(u) with the field theory 't Hooft coupling λ Y M , and write: (3.72) in terms of an unknown function f (λ). Then, we can rewrite (3.71) as: where in the second equality we have used the fact that dA/dλ = 1/β(λ) as follows from equation (2.27) and β(λ Y M ) indicates the beta function of the 't Hooft coupling λ Y M . This shows that the holographic operator defined in (3.69) with J inv = −A coincides with the RG-invariant gluon operator independently of the relation between the bulk field λ and the Yang-Mills coupling. The same cannot be said for the operator dual to λ −1 itself, which conicides with T rF 2 in the UV but it may deviate from it in the IR. We now procede to compute the effective potential for T . The RG-invariant operator T at the zero-derivative order is computed by the functional derivative of the homogeneous part of S (ren) , equation (3.8), with respect to J inv : where we have used equation (3.9) in the second equality.
If we recall the definition of the non-perturbative scale Λ (2.26), with our choice of the normalization, we see that the right hand side of (3.74) is nothing but D 0 N 2 c Λ 4 . It is convenient to fix the scheme such that D 0 = 1, so that the vacuum expectation value of T is simply given by: The effective potential for T , i.e. the zero-derivative part of the effective action, is given by: where the last term is the homogeneous part of S (ren) , given in equation (3.9). Using the full RG invariant coupling as a function of the vev T calculated from (3.74) and the background RG invariant couplingJ computed by inverting the definition of the non-perturbative scale the effective potential reads By construction, we have: which is consistent with (3.74). From the final result of (3.80), we can see that the effective potential is modelindependent and it has a universal form for arbitrary bulk scalar potential V (φ). The RG scale µ dependence in the effective potential is cancelled as well, so the form of Γ (0) [T ] is RG invariant, which is natural for an RG invariant operator. As before, going to second order in space-time derivatives and substituting (3.77) into the general formula (3.19), one can derive the two-derivative terms in the 1PI effective action, from which one is able to find out the canonically normalized operator T (c) using the kinetic term and to write the 1PI action in terms of the canonically normalized operator T (c) .
Here we will present the analytic results of the RG-invariant operator for the UV and IR, where one can expand the superpotential for small and large λ respectively.
In the UV, the RG invariant operator coincides with T rF 2 up to a numerical factor b 0 since A ∼ 1 b 0 λ from the UV expansion (A.1) and thus T ∼ b 0 2 G. We can obtain the effective action of T by simply substituting b 0 2 G by T , so the result is , (3.82) From the kinetic term, we define the canonically normalized operator T (c) as and the corresponding effective action has the same form as (3.60) The identification of the RG invariant operator T with the YM operator G in the UV explains the fact that the Γ U V [G] is also RG scale invariant.
In the IR, the effective action of the RG invariant operator T is different from that of the YM operator G. To compute the effective action, we first write the IR renormalized generating functional in a convenient form where (3.87) From the explicit form of the kinetic term, we determine the canonically normalized operator to be Finially, we derive the IR-limit of the effective action of the canonically normalized RG-invariant operator T (c) Looking at equations (3.84) and (3.89), notice that the UV and IR effective potentials of the canonically normalized vev T (c) have the same functional form: but the minima are different due to the change of α.

The bare gluoninc operators
In this section we will compute the holographic bare, regularized effective action. This is done by Legendre transformation of the bare action for the sources with an explicit UV boundary cut-off. We first calculate the effective action for the gluonic operator G = T rF 4 , then the one for the RG-invariant operator T = (−β λ ) T rF 2 /(2λ 2 ). Using the notation introduced in Subsection 3.2, the zero-derivative term in the bare action according to (3.7) reads (3.91) To compute the vev of bare gluon operator T rF 2 , as in Subection we take the functional derivative of the regularized action S (bare) with respect to the source J = (−2λ) −1 , To carry out the explicit computation of the effective action, we need to make a natural assumption that the background value of the coupling at the cut-off is very small, λ 1, (3.93) so the perturbative expansions around the UV fixed point is possible. As explained at the end of Subsection 3.2, the assumption (3.93) implies that the UV cut-off scale Λ U V is much large than the non-perturbative scale Λ, The equivalence (3.93) and (3.94) can be understood from the definition of the non-perturbative scale (2.30), where the physical scale µ replaced by the lattice cut-off = Λ −1 U V . Under the small λ assumption, the vev of G at the leading order becomes from which one can invert the relation between G (x) and λ to obtain the 't Hooft coupling as a function of the vev The background source at the cut-off at the leading order of small λ expansion can be calculated as well from which we can compute the sourcẽ . (3.99) Then we calculate the effective action by Legendre transform of the bare action (3.1) with respect to the sourceJ where higher order terms of G are neglected. The effective potential can be put in a form which makes manifest the violation of scale invariance: so the effective potential have a logarithmic scaling violation. The effective action of RG-invariant operator T has the same form since in the UV region the RG-invariant coupling is the same as the YM operator up to a numerical factor From the kinetic term, one can determine the canonically normalized operator to be which is the same for different operators due to the same form of kinetic terms when written as a function of φ. The expression for the zero-derivative term ( the potential term) depends on the specific definitions of its operators, but they coincide in the small λ limit. We have fixed the integration constant in such a way that the value of the canonically normalized operator at the minimum of the effective potential is As an illustration, we compute the effective potential of G using the simple potential (3.67), and we present the result in Figure 3. Now let us see explicitly how the renormalized v.e.v. T is obtained by subtracting the divergence pieces of the bare v.e.v. T by counter-terms. In terms of the cutoff , the bare vev at the minimum of the effective potential is where the finite terms start from C Λ 4 and C is the integration constant in the regular superpotential solution. The subleading divergent terms and the subleading finite terms are not written explicitly.
Since the counter-terms in the renormalized action have the same form as the bare Lagrangian but with different integration constants, the universal divergent terms in the bare v.e.v. will be subtracted. Of the remining terms, the ones that remain finite as we move the cut-off to infinity are: where C ct is the integration constant that governs the subleading part of the counterterm superpotential. The remaining subleading terms vanish as → 0. The

Effective potentials in the realistic IHQCD model
In this section we give a concrete example based on the Improved Holographic QCD model [6]. Our starting point is the full bulk potential of IHQCD: It is interesting to estimate the non-perturbative scale Λ, which we defined as (2.30), that corresponds to the fisical choice of units that matches real-life Yang-Mills theory. This can be computed, for example, by fixing the bulk solution and the asymptotic AdS scale in such a way that the lowest 0 ++ glueball mass coincides with the value calculated on the lattice 11 , m 0 ++ = 1710 M eV [20]. The resulting non-perturbative scale is Λ = 191 M eV (4.2) and it agrees with what is generally taken to be the scale of non-perturbative Yang-Mills theory, confirming that our definition of the non-perturbative scale can be matched concretely to the field theory result. Using the numerical superpotential solution, one can compute the zero-derivative term and the coefficients of the two-derivative terms in the renormalized generating functional. Then we compute the effective potential of the renormalized composite gluon operator G = T rF 2 and the renormalized RG-invariant operator T = −β(λ)T rF 2 /2λ 2 . The results are displayed in Figure 4 and Figure 5. In our numerical calculation, we have fixed the energy scale for G such that the background sources take simple values: The scheme dependent coefficients D 0 and D 1 have been chosen as in equation (3.58), but D 2 is chosen differently to set the vev of canonically normalized operator at the minimum to be N 1/2 c Λ. Of the curves shown in figures 4 and 5, only the blue one in figure 4 is independent of the choice of scale, as it corresponds to the effective potential for the un-normalized RG-invariant operator. Changing the choice of the reference scale affects the other curves in a way similar to that shown in figure 1. Notice that the canonically normalized operators T c and G c have the same v.e.v. This is because, as explained below eqaution (3.42), both v.e.v.'s are given in terms of the same function of the scalar field φ.

Conclusion
In this work we have shown, in a phenomenological gravity dual in five dimensions, how to compute the effective action, up to two derivatives, for the lowest dimension, scalar single trace operator in Yang-Mills theory: the dimension-four scalar glueball operator. We have found for the RG-invariant glueball operator a universal, simple analytic form for the potential, given in equation (3.80), which nicely incorporates the conformal anomaly. Notice that this form is not restricted to QCD-like theories, but to any holographic model driven by a single scalar. It would be interesting to extend this and the other calculations in this paper to the multi-field case and see whether this simple structure persists. Switching to canonical normalization the resulting potential ceases to be universal, because the effective kinetic term does depend on the specific bulk theory.
Beside presenting the calculational framework and providing model-independent results, we have also analyzed in detail the effective potentials arising in specific models which were proposed as realistic phenomenological gravity duals of Yang-Mills theory. These models are known to reproduce very accurately the static thermal properties of finite-temperature Yang-Mills theory [6], and it would be interesting to investigate whether this agreement extends to other static quantities like the gluonic effective potential. The main obstacle here is the fact that a reliable lattice computation of this quantity is hard to achieve, due to the amount of short-distance noise in this channel. However, the only source of scheme dependence is in the overall scale of the potential, not its shape: thus, if one could manage to subtract the UV contribution to the vacuum energy, the shape of the remaining effective potential should be completely fixed, and comparison with the holographic results presented here should be possible.
The effective actions we constructed encode the full quantum dynamics in cases when it is governed by a single operator of the theory, and mixing with other singletrace operators is weak. This is not necessarily the case for the true Yang-Mills theory, but it can be used as a first approximation, which can be in principle be checked, for example on the lattice.
Although it has the standard form of Lorentz-invariant kinetic plus potential terms, the effective action we obtained must not be interpreted as the action for a field describing physical particle excitations. Thus, small oscillations around the vacuum do not describe modes that have a direct particle interpretation 12 .
Rather, the effective action encodes the collective behavior of the full tower of physical particle modes (in this case, the tower of 0 ++ scalar glueballs) associated to the gauge-invariant operator in question. Our holographic models, like Yang-Mills theory, have an infinite discrete spectrum of states, whose masses m n are set by the non-perturbative scale Λ. These states are the eigenmodes of the linearized scalar bulk fluctuations, and they appear as poles in the two-point function of the dual operator O: Generically all low-lying states have comparable masses and decay constants, Thus, unlike the case of a (quasi)-free field, the dynamics around the vacuum cannot be associated to any one particular physical particle.
To illustrate this point further, let us consider O(x) as a small perturbation around the vacuum O , and let us expand the quantum effective action, of the general form (1.3), to quadratic order in O(x) as well as second order in momenta: In the above expression, Γ 0 is the vacuum energy, and Γ 2 (p) is the "self-energy," which we have written up to quadratic order in momenta, in terms of the functions V and G parametrizing (1.3), here evaluated on the vacuum O . Since Γ[O] is the Legendre transform of the generating functional of connected correlators, we have the standard relation between the two-point function (5.1) and the self-energy, .
Thus, using (5.1) for the right hand side and expanding it to second order in momenta we can find relations, in the form of sum rules, between the spectral data m n , f n and the "mass" and "kinetic" term appearing in the effective action: In these expressions, C 0 and C 2 are the coefficients of the constant and p 2 term in expansion of the analytic part of the two-point function (5.1). These correspond to contact terms which have UV divergences and are subject to renormalization. Thus, their finite part is scheme dependent, and this must match the scheme dependence of the coefficients of the effective action on the left hand side of (5.5). It would be interesting to check these sum rules explicitly in models with a known spectrum. We leave a more detailed investigation of this problem, and of the precise way scheme dependence enters into the sum rules, for future work.
From equation (5.5) it is clear that the second derivative of the potential cannot be associated to any one particle mass, but it is given by a collective effect of all physical particle excitations. This is to be contrasted with the case of the potential in a weakly coupled field theory.
One interesting exception is when one of the modes is much lighter than the others, i.e. when the theory has a light dilaton-like particle excitation: in this case the first term in the sums dominates, the dynamics is dominated by the light mode, and the field O behaves approximately like a free field whose associated particle is the light mode. In these special cases, the effective action we have constructed can be thought of as the effective action for the composite light modes, in the spirit of [13]. However, in holographic models this situation seems to be non-generic, and requiring fine tuning [12].
In the generic case, on the other hand, assuming O is the operator driving the dynamics, the effective actions computed in this paper will describe the vacuum structure and evolution purely in classical terms, as a collective behavior of O seen as a classical field, and there will be no narrow-width particle-like excitations. The framework we have developed thus offers an intuitive Lagrangian tool to describe the dynamics of condensates which are not necessarily associated to particles, and can find many uses in holographic phenomenology.

Acknowledgements
We would like to thank M. Panero and A. Patella for discussion. This work was supported in part by European Union's Seventh Framework Programme under grant agreements (FP7-REGPOT-2012-2013-1) no 316165, PIF-GA-2011-300984, the EU program "Thales" MIS 375734, by the European Commission under the ERC Advanced Grant BSMOXFORD 228169 and was also co-financed by the European Union (European Social Fund, ESF) and Greek national funds through the Operational Program "Education and Lifelong Learning" of the National Strategic Reference Framework (NSRF) under "Funding of proposals that have received a positive evaluation in the 3rd and 4th Call of ERC Grant Schemes". We also thank the ESF network Holograv for partial support. From dimensional analysis, one can see the gluon operators have the correct scaling dimensions: G ∼ µ 4 and G (c) ∼ µ.
In terms of the canonically normalized operator G (c) , the 1PI effective action of the vev T rF 2 in the UV reads: (A.12) This expression for the one-loop effective action realizes the field theory expectation based on the conformal anomaly (see e.g. [19]).

A.2 1PI effective action in the IR
Using the IR solution of the superpotential (2.20), the function A(φ) in the renormalized generating functional action (3.8) can be written as a function of λ where the subleading terms in the λ → ∞ limit will be neglected in the following discussion.
Using the critical value Q = 2 3 , the expotential of A is simplified into e A = λ −Q (ln λ) P 2 . (A.14) Substituting e A by its IR asymptotic form (A.14), the renormalized generating functional (3.8) becomes:  (1) 0 W 2 for large φ to simplify the coefficient of the kinetic term. The IR subleading terms have been neglected. One can see that the contribution from G (1) 0 is important although its UV contribution is negligible. Similar to the UV computations, we have introduced the energy scale µ as the scale factor of the flat induced metric γ µν = µ 2 η µν .
At the zero-derivative order, the vev G is derived from the functional derivative of the renormalized generating functional (3.8) with respect to J = (−4λ) −1 : We can express the coupling as a function of the vev by inverting the relation between G and J = (−2λ) −1 .

(A.18)
We can now calculate the effective action by Legendre transforming S (ren) with respect to the fluctuations of the coupling The effective action up to two-derivative order is computed using (A.2) Γ[G, µ,λ(µ, Λ)] = GJ − S (ren) 20) where in the last line we have neglected the subleading terms in the limit G/µ → ∞ in the potential and we can see that the effective potential has a linear dependence on the vev G with a slope (4λ) −1 .
Then we need to express the background coupling constant in terms of the nonperturbative scale (2.30) From the kinetic term, one determines the canonically normalized operator as where G 0 (φ) is defined in (3.12) and the subscript IR indicates it is evaluated in the IR.