Effective holographic models for QCD: glueball spectrum and trace anomaly

We investigate effective holographic models for QCD arising from five dimensional Dilaton-Gravity. The models are characterized by a dilaton with a mass term in the UV, dual to a CFT deformation by a relevant operator, and quadratic in the IR. The UV constraint leads to the explicit breaking of conformal symmetry whereas the IR constraint guarantees linear confinement. We propose semi-analytic interpolations between the UV and the IR and obtain a spectrum for scalar and tensor glueballs consistent with lattice QCD data. We use the glueball spectrum as a physical constraint to find the evolution of the model parameters as the mass term goes to zero. Finally, we reproduce the universal result for the trace anomaly of deformed CFTs and propose a dictionary between this result and the QCD trace anomaly. A nontrivial consequence of this dictionary is the emergence of a $\beta$ function similar to the two-loop perturbative QCD result.


Introduction
Quantum Chromodynamics (QCD) is probably the most striking example of conformal symmetry breaking in a Quantum Field Theory. In QCD conformal symmetry is broken already in the UV by a negative β function (asymptotic freedom) that, at the same time, leads to strong coupling and confinement in the IR. Despite the huge success of perturbative QCD in describing hard scattering processes, basic QCD features in the IR, such as the hadronic spectrum or chiral symmetry breaking, require the development of nonperturbative methods. The most well established non-perturbative approach consists of Monte-Carlo simulations for QCD on a lattice. The so-called lattice QCD is very successful in describing static properties such as the hadronic spectrum and thermodynamic properties of the quark-gluon plasma. However, real-time dynamics usually demand the development of other non-perturbative methods.
Holographic QCD provides a non-perturbative description of real-time QCD dynamics in terms of the dynamics of a five dimensional gravitational (string) theory. In the so-called bottom-up approach, it is used the dictionary arising from the AdS/CFT correspondence to build a set of five dimensional fields dual to the QCD operators responsible for describing the QCD vacuum. In the large-N c limit one focuses on the stress-energy tensor T µν of the gluon field, as well as on the scalar operator Tr F 2 responsible for the gluon condensate and the QCD scale anomaly, i.e. T µ µ ∼ Tr F 2 [1]. In holography the operators T µν and Tr F 2 couple to a five dimensional metric and scalar field (the dilaton), respectively. The dynamics of the metric and dilaton fields is determined by the equations of the Dilaton-Gravity theory, and the dilaton potential contains a negative cosmological constant leading to AdS geometry in the UV. Since Lorentz invariance is not broken by the QCD vacuum, in holographic QCD one takes a conformally flat metric with a warp factor A depending solely on the radial coordinate z. Then a non-constant dilaton Φ(z) leads to a deformation of the AdS spacetime geometry. This is the holographic realization of conformal symmetry breaking in large-N c QCD. Assuming that Φ(z) couples directly to the operator Tr F 2 at the boundary one would expect, according to the AdS/CFT dictionary, Φ(z) to behave as φ 0 + G z 4 near the boundary. This assumption, however, was shown not to be consistent with the glueball spectrum [2], because in that case conformal symmetry is spontaneously broken, leading to a zero mode in the spectrum (a Nambu-Goldstone boson).
There are two possible solutions to this problem. The first one consists of making the dilaton potential compatible with asymptotic freedom in the UV, as advocated in [2]. This scenario was realized in the so-called Improved Holographic QCD models [3,4], proposed by Gursoy, Kiritsis and Nitti. These authors also found the right physical constraint for the dilaton in the IR: a confining background, which corresponds to a quadratic dilaton for large z and leads to an approximate linear glueball spectrum. Interestingly, that behaviour had already been anticipated in the soft-wall model [5] from the analysis of the meson spectrum. The second solution consists of introducing a mass term for the dilaton potential in the UV. In this case the near-boundary behaviour of the dilaton field becomes φ 0 z + G z 4− , with related to the dilaton mass M by M 2 = − (4 − ). This UV asymptotics was proposed by Gubser, Nellore, Pufu and Rocha in [6,7], within the context of finite-temperature holographic QCD as a model for describing the equation of state of a non-conformal plasma through a five-dimensional black hole. In [6,7] the parameter was interpreted in terms of the anomalous dimension of the operator Tr F 2 .
Inspired by the seminal works [3,4] and [6,7], we investigate in this paper a family of holographic QCD backgrounds where the IR is driven by linear confinement and the UV, although asymptotically AdS, is deformed by a dilaton field with nonzero mass. We interpret the dilaton mass term in the UV as the holographic description of a CFT deformation δL = φ 0 O, where O is a relevant operator and φ 0 is the corresponding coupling. This interpretation was advocated in [6,7] and is consistent with previous studies in holographic renormalization [8,9]. The main motivation for this work is to understand how holography realizes conformal symmetry breaking in the UV without introducing explicitly the β function of large-N c QCD. Our guide in this investigation will be the spectrum of scalar and tensor glueballs, because it will allow us to fix the source and the VEV coefficients in the near-boundary expansion. 1 As in refs. [6,7], we assume that the CFT deformation takes place at a UV energy scale E * . However, although 4 − is indeed the conformal dimension of the operator O (responsible for the CFT deformation), the relation between O and the QCD operator Tr F 2 is not direct. Therefore, we will not interpret the parameter as the anomalous dimension an of the operator Tr F 2 , as was advocated in [6,7]. We will, however, propose a relation between these two quantities.
We show in this work that for arbitrary values of , in the range 0.001 < < 0.1, it is always possible to reproduce the spectrum of scalar and tensor glueballs obtained in lattice QCD. This is achieved by using the first two scalar glueball masses as a physical criterion to fix the source and VEV coefficients, φ 0 and G, respectively, for each value of . We find that the evolution of these parameters as functions of admits simple fits that allow us to predict their behaviour in the → 0 limit (where the dilaton becomes massless). On the other hand, from the analysis of the vacuum energy density T 00 and the VEV of O we calculate the corresponding trace anomaly T µ µ = − φ 0 O , which is consistent with the general result of deformed CFTs [9,10]. We will suggest a reinterpretation of this result in terms of the QCD trace anomaly, which in turn will suggest a dictionary for the parameters φ 0 and .
Our approach is quite different from the one considered in [3,4], where the 4-d theory does not have a UV cut-off and the dilaton potential is built to reproduce the two-loop perturbative β function of large-N c QCD (asymptotic freedom). It is not clear, however, how the holographic map between the energy E and the warp factor A(z), proposed in [3,4], should be modified in our case in order to accommodate the energy cut-off E * . In recent works [15,16], a massive term for the dilaton in the UV was interpreted as the dual of a CFT deformation by a nearly marginal operator and the corresponding RG flow was investigated (see also [17]). Our results, however, support the interpretation of the CFT deformation δL = φ 0 O in terms of the large-N c Yang-Mills Lagrangian. In some sense this is a reinterpretation of the RG flow in holographic QCD where now the operator Tr F 2 is interpreted as the source of a nontrivial β-function. This may also shed some light in the origin of the dilaton potentials considered in holographic QCD models without relying on a stringy top-down approach.
This paper is organized as follows. In section 2 we review the holographic QCD models arising from five-dimensional Dilaton-Gravity theory, with focus on the linearised equations that lead to the glueball spectrum. Then, in section 3, we describe the physical constraints in the UV and IR, compatible with explicit conformal symmetry breaking and linear confinement, and present two interpolations where the dilaton admits an analytic form. In section 4 we find a spectrum for scalar and tensor glueballs compatible with lattice QCD data and describe how the model parameters evolve with the conformal dimension . In section 5 we calculate the trace anomaly for our model, which agrees with the general expectation of deformed CFTs and compare this result with the QCD scale anomaly. We finish this paper with our conclusions and two appendices. Appendix A describes massless scalar modes in holographic QCD. In appendix B we present two holographic QCD models where the warp factor admits an analytic form.

Review of holographic QCD
In this section we review the holographic QCD backgrounds (HQCD) in a bottom-up approach, which makes use of the field/operator correspondence unveiled by the AdS/CFT correspondence. In order to describe the large-N c QCD vacuum the focus is on the stressenergy tensor T µν as well as the Yang-Mills Lagrangian operator Tr F 2 . The former couples to a five dimensional metric g mn and the latter couples to the scalar dilaton field Φ. Then the natural five dimensional framework is the Dilaton-Gravity theory, where besides the Einstein-Hilbert action one considers a dilaton kinetic term and a dilaton potential.
First we will exploit the fact that HQCD backgrounds are conformally flat and map the perturbed metric to linearised gravity around the Minkowski spacetime. Then we will briefly review the general features of the models proposed in [3,4], also known as Improved Holographic QCD. We finish the section with a full description of the linearised Dilaton-Gravity equations leading to the glueball spectrum.

Ricci tensor in HQCD backgrounds
Consider a Weyl transformation for a 5-d metric: We take the transformation (2.1) as a field redefinition for the metric, or as an ansatz for the background spacetime 2 . In holography the 5-d coordinates x m of the bulk spacetime decompose as (x µ , z) where x µ are the 4-d coordinates associated with the field theory at the boundary and z is the bulk radial coordinate.
The metrics g mn andḡ mn include also the fluctuations around the unperturbed background, and so they admit the expansions g mn =ḡ (0) mn +ḡ (1) mn + . . . , g mn = g (0) mn + g (1) mn + . . . , (2.2) where we takeḡ (0) mn as the metric associated with a reference background. Analogously, the Christoffel symbols Γ p mn = 1 2 g pq (∂ m g qn + ∂ n g qm − ∂ q g mn ) and the Ricci tensor The Ricci tensor transforms under (2.1) as In HQCD we are interested in the case where the reference background is flat, i.e.
and we take the warp factor A as a function of the radial coordinate only, i.e. A = A(z) so that 4-d Poincaré symmetry, associated with the coordinates x µ , is preserved. Then at 0th order the Ricci tensor of the dual metric takes the form where means d/dz. Projecting out the Ricci tensor we obtain the components and the 0th order Ricci scalar R ≡ g mn R mn takes the form On the other hand, the reference metric at 1st order is that of linearised gravity around flat space, i.e.ḡ (1) where the parentheses around the indices denote symmetrization, V (mn) ≡ (V mn + V nm )/2, and the indices of h mn are raised (lowered) using the Minkowski metric η mn (η mn ). Ex-panding both sides of (2.4), we find that the 1st order perturbations of the Ricci tensor can be written as Decomposing the tensor of linearised gravity h mn as (h zz , h zµ , h µν ) and defining h zz ≡ 2φ and h zµ ≡ A µ , we can project out the Ricci tensor (2.10) and find the following components: where ≡ ∂ µ ∂ µ is the d'Alembertian operator in the boundary spacetime, the scalar h is defined by the trace h ≡ h µ µ and F µν ≡ ∂ µ A ν − ∂ ν A µ is the field strength associated with the vector A µ .

HQCD backgrounds from Dilaton-Gravity
The goal of holographic QCD is to find the gravity (string) dual of QCD in the large-N c limit. This is motivated by the 't Hooft planar limit [18] and the AdS/CFT correspondence [19][20][21]. QCD is, on the one hand, well approximated by a CFT in the UV and, on the other hand, confining in the IR. These facts suggest that the holographic dual spacetime should be AdS near the boundary. Far from the boundary, it should be such that dual probe fields (and strings) living in that 5-d background reproduce confinement and the 4-d hadronic physics. In the pioneer work of [3,4], a very general family of HQCD backgrounds was proposed and, based on the work of [22], a universal IR criterion for confinement was found. Moreover, the requirement of linear Regge trajectories led the authors of [4] to conclude that a quadratic dilaton field was necessary in the IR. This supports the early work of [5], where a quadratic dilaton was proposed on the basis of the meson spectrum, the so-called soft-wall model.
In this subsection, we briefly review the HQCD backgrounds proposed in [3,4] focusing on the IR physics. In the next subsection, we will describe the scalar and gravitational perturbations that lead to the Schrödinger equations associated with the glueball spectrum.
In the HQCD approach, we start with a 5-d Dilaton-Gravity action of the form where M p is the 5-d Planck scale, N c is the number of colors, and the dilaton Lagrangian has a kinetic term and a potential, Variating the action (2.12) with respect to Φ and the metric g mn , we obtain the Dilaton-Gravity equations 14) where ∇ 2 is the Laplacian operator 3 and we have defined an energy-momentum tensor T mn for the dilaton field: It is also convenient to write the Einstein equations (2.14) in the Ricci form: (2.17) The HQCD backgrounds correspond to solutions for the Dilaton-Gravity equations of the form The warp factor A(z) and dilaton Φ(z) are usually mapped to the energy scale and coupling of the dual 4-d theory. Using (2.7) and the definition of the scalar Laplacian ∇ 2 , the Dilaton-Gravity equations (2.15) and (2.17) take the form The last equation in (2.19) can be obtained from the first two, which in turn can be rewritten as (2.20) At this point it is very convenient to define the quantity

21)
3 The Laplacian operator ∇ 2 applied to a scalar function f is given by so that the first equation in (2.20) takes a linear form in ζ: The Schrödinger form of this equation is useful to understand the AdS deformation due to a non-constant dilaton, which is the dual of a conformal symmetry breaking. In the case of a constant dilaton, the interesting solution for the holography is ζ = z/ , corresponding to the usual AdS spacetime with curvature radius . In this paper we will describe how the presence of dilaton mass, associated with a CFT deformation, leads to an explicit breaking of conformal symmetry and gives a reasonable glueball spectrum. Note from equation (2.22) that ζ ≥ 0 for all values of z. AdS asymptotics implies that ζ (0) = 1/ so we conclude that ζ ≥ 1/ . 4 Following ref. [4] we write the equations (2.20) as a system of first order differential equations where W = W (Φ) is the superpotential associated with the Dilaton-Gravity dynamics. Another useful quantity is the field X defined by the relations This field can be interpreted as a bulk beta function X ∼ β Φ describing the evolution of the dilaton Φ with the warp factor A. In the next subsection we will describe how this field appears in the Schrödinger equation associated with scalar glueballs. In [3,4] the authors proposed a dictionary that maps the bulk field X to the β function of the 4-d dual theory. In our work the CFT deformation in the UV implies the existence of a cut-off E * in the energy scale of the 4-d theory. This feature suggests a departure from the map proposed in [3,4] so we take X as a pure bulk field. In particular, we will see that while X goes to zero as we approach the boundary the dual β function is still finite.
We finish this subsection describing the confining constraint found in [4]. The discussion takes place in the string frame, where the metric is given by 25) and the string-frame warp factor is related to the Einstein-frame warp factor by Consider a static string living in the spacetime (2.25), with endpoints attached to the boundary and separated by a distance L in one of the boundary directions. As shown in [22], inspired by [23], this problem maps to a rectangular Wilson loop describing in the large L limit the potential energy of a heavy quark-antiquark pair. Solving the Nambu-Goto equations one finds that in the large L limit the energy of the static string takes the form E(L) = µf (z * )L + . . . , (2.27) where µ is the fundamental string tension, f (z) = exp(2A s ), and z * is the point where f (z) has a minimum and the dots represent subleading terms in the large L limit. Following ref. [22], the energy (2.27) maps to the quark-antiquark potential and confinement is achieved for f (z * ) > 0. In this case, the quantity µf (z * ) is identified with the confining string tension σ.
Thus we conclude that confining backgrounds are those where the function f (z) = exp(2A s ) has a non-zero minimum. Since we always consider backgrounds that are asymptotically AdS we have that f (z) → ∞ in the UV (z → 0). We are interested in backgrounds where 0 < z < ∞ so we conclude that in the IR f (z → ∞) > 0 to guarantee confinement. Taking a power ansatz for the dilaton Φ(z) = z α we find from (2.20) that at large z Then the confinement criterion becomes the condition α ≥ 1. From (2.8) and (2.28) we find that the condition α ≥ 1 implies the existence of a curvature singularity at z → ∞. Interestingly, a WKB analysis of the glueball spectrum [4] leads to the stronger restriction α = 2 that corresponds to asymptotically linear Regge trajectories m 2 n ∼ n. In this work we will take a quadratic dilaton Φ(z) = z 2 in the IR to guarantee confinement and an approximate linear glueball spectrum. The UV, on the other hand, will differ significantly from the proposal of [3,4] where instead of imposing asymptotic freedom we will consider a CFT deformation, inspired by the work of [6,7]. But first we will finish this section by reviewing below how the Schrödinger equations, that determine the glueball spectrum, arise from the linearised Dilaton-Gravity equations.

Linearised Dilaton-Gravity equations
The linearised version of the Dilaton-Gravity equations are obtained by expanding at first order both sides of (2.15) and (2.17), with Φ → Φ + χ and g mn → e 2A (η mn + h mn ), where χ and h mn are first-order perturbations in the dilaton and the reference background metric, respectively. The resulting equations take the form mn is given by (2.11), and recall that h zz = 2φ, h zµ = A µ and h µ µ = h. Taking the components (zz, zµ, µν) of the linearised Einstein equations (2.30), the system (2.30)-(2.31) becomes where we have used the result (2.11) for the Ricci tensor and we have also used the following background relations As explained in [24], the next step is to decompose the four-vector A µ and the symmetric tensor h µν into irreducible representations of the Lorentz group, i.e, where A T µ and V T µ are divergenceless vectors, h T T µν is a traceless and divergenceless tensor and W, E, ψ are Lorentz-scalars. Applying the decomposition (2.38) into the Dilaton-Gravity equations (2.33)-(2.36), we find one tensorial equation and five scalar equations The tensorial equation ( This equation can be combined with (2.46) to get rid of the term (W − E ) and to find where X was defined in (2.24). Using equation (2.44) to replace φ in terms of ψ and χ and the background relations (2.20) and (2.37), we arrive at the decoupled equation where the field ξ is defined by The solutions of equation (2.49) lead to the spectrum of spin 0 glueballs.

Schrödinger like equation -scalar sector
It is possible to rewrite equation (2.49) in a Schrödinger like form. To do so, we define an auxiliary function B s as Substituting ξ = e −Bs ψ s and introducing the Fourier transform ( → m 2 s ) in equation where the potential is defined as

Schrödinger like equation -tensor sector
Following the same procedure as above in the tensor sector we use the auxiliary function to rewrite (2.39) in the Schrödinger like form where the potential is defined as Interestingly, the difference between the spin 0 and spin 2 sectors lies in the term B s −B t = log(X). This is an important feature when calculating the glueball spectrum that explains the non-degeneracy of scalar and tensor glueballs.

Effective holographic QCD
As anticipated in the previous sections, our work will explore the idea of a massive term for the dilaton in the UV as the dual of a CFT deformation δL = φ 0 O. The coupling φ 0 and the operator O have conformal dimensions and 4 − , respectively. The latter will be related to the QCD operator Tr F 2 . The idea of a CFT relevant deformation was proposed in [6,7] when constructing a holographic model for QCD at finite temperature. Since the CFT deformation takes place at a particular energy scale E * , which becomes an upper energy bound for the 4-d theory, we dub this approach effective holographic QCD.
In this work we will show that this type of UV asymptotics, at zero temperature, is compatible with (explicit) conformal symmetry breaking, confinement and the glueball spectrum. To achieve this we also constrain the IR in the way proposed in [3,4], namely by considering a quadratic dilaton. Below we describe in more detail the universal UV and IR asymptotic behaviour for the family of HQCD backgrounds considered in this work. We also describe specific models that interpolate smoothly between the UV and IR asymptotics. Then in the next section we will calculate the glueball spectrum and investigate how the model parameters evolve with the conformal dimension .

CFT deformation in the UV
In the pioneer work [2], Csaki and Reece identified a dynamical dilaton as the fivedimensional scalar field dual to the operator Tr F 2 . They initially considered a massless dilaton and obtained a nonzero gluon condensate. However, as found in [2], a massless dilaton leads to a Nambu-Goldstone boson in the spectrum of scalar glueballs, indicating that conformal symmetry was spontaneously broken. It is important to remark, however, that the background of [2] requires additional boundary conditions at the singularity, which brings some ambiguities [4]. As a mechanism for explicit conformal symmetry breaking, the authors of [2] proposed the holographic implementation of asymptotic freedom in the UV. This was correctly implemented in the IHQCD background developed by Gursoy, Kiritsis and Nitti [3,4]. The background of [3,4] is also consistent with linear confinement in the IR and does not require additional boundary conditions at the singularity.
As explained above, instead of implementing asymptotic freedom in the UV, we are interested in explicitly breaking conformal invariance through a UV mass term for the dilaton. According to the AdS/CFT dictionary, the mass M of a 5-d dilaton, responsible for deforming AdS space, is related to the conformal dimension ∆ of a dual scalar operator O, responsible for deforming the 4-d CFT, by the equation M 2 = ∆(∆ − 4). Then by considering a relevant operator with dimension ∆ = 4 − , we end up with a nonzero mass for the dilaton. We will later relate this operator to the QCD operator Tr F 2 and investigate the connection between the trace anomaly of deformed CFTs and the QCD trace anomaly. This in turn will shed some light on the role played by the conformal dimension in effective holographic QCD.
Let us first consider an expansion of the dilaton potential around the UV minimum Φ = 0, which includes a constant term, associated with a negative cosmological contant, and a nonzero mass term: where the ellipses denote higher powers of Φ. On basis of equations (2.19), one finds that the constant term in (3.1) leads to the AdS asymptotics, with an AdS radius = 1, whereas the mass term implies the following near boundary behaviour for the dilaton: where ∆ + = ∆ = 4 − and ∆ − = 4 − ∆ + = , both related to the 5-d mass by M 2 = −∆ + ∆ − . Following the AdS/CFT dictionary, we will interpret the coefficient φ 0 as the source for the dual operator O. The coefficient G will be related to the VEV of O. We also remind the reader that the CFT deformation δL = φ 0 O takes place at some cut-off energy E * .
Considering the perturbative expansion for Φ, given by (3.2), we find from equation (2.22) the asymptotic expansions for ζ(z) and the warp factor A(z) = − log ζ(z): If instead of a mass term for the dilaton, we followed the prescription of [3,4] and imposed asymptotic freedom, we would obtain a UV asymptotics involving logarithmic terms to be consistent with the logarithmic dependence of the 't Hooft coupling with the energy.
Since we know the asymptotic behaviour of the warp factor (3.3) we are also able to find, from the second equation in (2.23), the small z behaviour of the superpotential, Alternatively, we can solve the differential equation (2.23) for the superpotential W (Φ) for a dilaton potential V (Φ) given by (3.1) and find [16,17] W ± (Φ) = 9 4 + ∆ ± 2 Φ 2 + . . . , (3.5) which is in agreement with equation (3.4). From the above results it is easy to find the asymptotic expansion for the field X(z), defined in (2.24). The expansion takes the form The results above also tell us how the metric behaves near the boundary, We finish this subsection writing the asymptotic expansion of the dilaton (3.2) in a form that will be useful when implementing the numerical procedure, where the parametersφ 0 and Λ are related to φ 0 and G by The parameter Λ has conformal dimension 1 and plays a role similar to Λ QCD whereas the parameterφ 0 is the massless version of the coupling φ 0 . For all practical purposesφ 0 , Λ and will be the relevant parameters of the model. We will see later that, for fixed , the parametersφ 0 and Λ can be fit in order to reproduce the glueball spectrum.

Confinement in the IR
In ref. [4], Gursoy, Kiritsis and Nitti did a careful analysis that is universal for holographic QCD backgrounds arising from a Dilaton-Gravity theory. Specifically, considering the general confinement criterion of [22] and also a WKB analysis for the glueball spectrum, they found that a quadratic dilaton in the IR guarantees confinement and an approximate linear spectrum for glueballs. Remarkably, this quadratic dilaton had been already proposed in the phenomenological soft-wall model [5], in order to arrive at a linear spectrum for mesons. Interestingly, a quadratic dilaton in the IR also provides the T 2 correction to the stress tensor trace anomaly of a deconfined plasma [25].
Motivated by the results of [4], we consider in this work the following dilaton asymptotic behaviour at large z: where the dots indicate terms that depend on negative powers of z. Using the IR asymptotic relation (2.28) between A(z) and Φ(z), we construct the asymptotic forms for the warp factor A(z) and the function ζ(z): With this information at hand, we use the second equation in (2.23) to write down the superpotential, We get the asymptotic expansion of X using equations (3.10) and (3.11) and the definition (2.24): Substituting this expression for X(Φ) in (2.24) and integrating the resulting equation in Φ, we obtain the asymptotic behaviour for the superpotential in terms of Φ, which is consistent with (3.12). Similarly, the asymptotic expression for the dilaton potential in the radial coordinate z can be found from equations (2.20) and (3.11), For completeness, we write down the dilaton potential as a function of Φ. Substituting equation (3.14) into the differential equation (2.23) and taking the leading term, we get The results (3.12) and (3.15) satisfy the general criteria presented in [3,4] to guarantee linear confinement in the IR. It is interesting to point out the difference between this IR asymptotics and the one considered in [6,7]. In that case the ratio ∂ Φ V (Φ)/V (Φ) is a constant because the potential goes like exp(γΦ), where γ is a constant. Here we have considered the IR asymptotics of [3,4], where the ratio ∂ Φ V (Φ)/V (Φ) has a subleading term that decreases as 1/z 2 because the dilaton is quadratic in the IR. From the result (3.11), we see that the metric shrinks to zero at large z as As explained above, this leads to a curvature singularity at z → ∞, i.e. a divergent Ricci scalar.
The above results were obtained in the Einstein frame, where the warp factor decreases monotonically. If we calculated the warp factor in the string frame, A s (z), we would see that it has a minimum, associated with the fundamental string tension [4].
Again, we write the IR dilaton asymptotics (3.10) in a convenient form Note that we are using the same coefficient Λ that appeared already in the UV expansion. This implies a relation between the IR coefficient C at large z and the UV coefficient G at small z. In the next section, we will show that the parameter Λ will be responsible for fixing the scale of the glueball masses when comparing the numerical results against lattice data.

UV/IR interpolation
In the pioneer soft-wall model [5], a quadratic dilaton was introduced by hand to get the desired behaviour in the dual QCD-like theory, namely, the Regge-like behaviour m 2 ∝ n. As explained before, starting from [2] there have been interesting proposals for holographic QCD considering a dilaton field dynamically coupled to the metric. This in turn leads to a nonzero gluon condensate and confinement. A particularly interesting proposal was considered recently in [26,27] where an analytic function was used to interpolate the dilaton between a quartic form in the UV and a quadratic form in the IR. However, as explained before, a quartic dilaton in the UV necessary leads to an unacceptable massless mode in the scalar sector of glueballs [2]. 5 In section 2.3 we described the process of linearising the Dilaton-Gravity equations to arrive at the equations governing the dynamics of the scalar and tensor glueballs. A careful analysis of these equations suggest two possible solutions for the massless mode problem. The first solution, originally proposed in [2] and beautifully realised in [3,4], consists of introducing asymptotic freedom in the UV. However, the price to be paid when introducing asymptotic freedom is the presence of logarithmic corrections in the warp factor and the dilaton which make the AdS/CFT dictionary more involved [28,29]. The second solution, considered in this work, is implementing a CFT deformation of the form δL = φ 0 O, being O a relevant operator dual to the dilaton. The conformal dimension of O was set to ∆ = 4 − in equation (3.2), with small. Mapping this operator to the QCD operator Tr F 2 signalizes the presence of a UV cut-off E * in the QCD-like theory, as advocated in [6,7].
The UV asymptotics corresponding to the CFT deformation was described in subsection 3.1 whereas the IR asymptotics leading to linear confinement was described in subsection 3.2. In particular, the UV and IR asymptotics for the dilaton were given in (3.8) and (3.18) respectively. There are two immediate options for interpolating between the UV and IR asymptotics. We can define an analytic function for the dilaton that interpolates from the UV to the IR and solve numerically for the warp factor and the potential V (z). The models arising in this approach will be called models A. The second possibility is to interpolate the warp factor between the two regimes, so that we get numerically the dilaton and the potential V (z). The models in this second approach will be called models B. In the following we present and analyse the first case (models A) and leave the analysis of models B for the appendix B.
In section 4 we will show how these effective holographic QCD models (models A and B) lead to a realistic spectrum for scalar and tensor glueballs. In section 5 we will take advantage of the fact that the effective holographic QCD approach allows the use of the standard AdS/CFT dictionary and calculate the VEV of the operator O. We will relate this VEV to the gluon condensate Tr F 2 and discuss the trace anomaly of deformed CFTs in connection with the QCD scale anomaly. A general discussion of the massless mode and its resolution is done in appendix A.

Models A: analytic form for the dilaton field
As explained above, in the models A we interpolate the dilaton field, from the UV asymptotics (3.8) to the IR asymptotics (3.18), in order to describe a CFT deformation in the UV and confinement in the IR. Among the many possibilities for interpolating the dilaton between the UV and IR asymptotics, we choose two of them.
The first interpolation (model A1) is constructed in terms of powers of the holographic coordinate: where the parametersφ 0 , Λ, and were already defined below equation (3.8). This is the simplest way of interpolating from the UV to the IR.
The second interpolation (model A2) introduces a hyperbolic tangent function, It is easy to see that both equations (3.19) and (3.20) recover the previous asymptotic expansions in the UV and IR, equations (3.8) and (3.18), respectively. The warp factor, on the other hand, is obtained by solving the first differential equation in (2.20). As explained previously, the dilaton field Φ(z) and warp factor A(z) will depend only on the parametersφ 0 , Λ and . The parameter Λ is used to fix the energy scale while the value ofφ 0 plays the role of a dimensionless coupling. Our numerical strategy will be to fix the conformal dimension and fit the parametersφ 0 and Λ using the masses of the first two scalar glueballs (taken from lattice QCD). This analysis will be developed in section 4.

Numerical analysis of the background
Having specified the models A, where the dilaton is an analytic function, we can solve numerically the Dilaton-Gravity equations (2.20), with the appropriate boundary conditions, and explore the evolution of the geometric quantities such as the warp factor A(z), the field X(z) and the superpotential W (z). The parameters that we use to get the results and plot the figures in this section are presented in Table 1 and will be justified in section 4.4. Our goal here is to show the nonsingular behaviour of these quantities. The dilaton field Φ(z), dual to the relevant operator O, is shown in Fig. 1. The effect of the conformal dimension is evident near the boundary where the dilaton field goes as ∼ φ 0 z (see the box in the figure), with φ 0 =φ 0 Λ . This is the dominant term responsible for the explicit breaking of conformal invariance. Figure 1 also shows the asymptotic behaviour of the dilaton ∼ Λ 2 z 2 in the IR, responsible for confinement and a linear spectrum. The difference between the models A1 and A2 lies on the values of the parametersφ 0 and Λ, in addition to the interpolation form.  On the left panel of Fig. 2, we plot the numerical solution for the Einstein-frame warp factor A(z) in models A1 and A2. These results are consistent with the UV and IR asymptotics, given by equations (3.3) and (3.11), respectively. The difference between models A1 and A2 lies in the region for large z. As we shall see in section 4, the effect of this difference is realized in the glueball spectrum. The right panel in Fig. 2 shows the string-frame warp factor A s (z), obtained from equation (2.26). This function has a minimum at some z = z * , which is consistent with the confinement criterion described in subsection 2.2.
Another important geometric quantity is the field X(z), defined in equation (2.24). As shown on the left panel in Fig. 3, this quantity has the same asymptotic behaviour for both models A1 and A2, and the relevant difference lies in the intermediate region. The presence of confinement in the IR is consistent with X(z) approaching a constant value for  Table 1.   Table 1. large z. As discussed in [15,16], the field X can be interpreted as a bulk effective β-function associated with the 4-d RG-flow driven by the operator O.
The breaking of conformal invariance can be appreciated by looking at the superpotential W (z), obtained from the second equation in (2.23). As a consequence of the CFT deformation in the UV, the superpotential raises rapidly with the coordinate z, as shown on the right panel in Fig. 3. This behaviour was previously noticed for the case of nearly marginal operators [15]. In contrast with X(z), the difference in W (z) between the models A1 and A2 lies in the IR. Those differences play a role in the spectrum of glueballs, that will be calculated in section 4.

The vacuum energy density
A very important quantity in holographic QCD is the vacuum energy density T 00 . In the absence of a CFT deformation, we expect T 00 to vanish since the 4-d theory lives in Minkowski spacetime. 6 In our case, the CFT deformation δL = φ 0 O will generate a nontrivial negative T 00 that will be interpreted as the QCD vacuum energy density.
In holography the vacuum energy of a 4-d theory corresponds to the on-shell Euclidean action associated with the 5-d gravitational background. In Dilaton-Gravity the action is given by where and is the Gibbons-Hawking boundary term. In holographic renormalization one first defines the boundary at z = z 0 and takes the limit z 0 → 0 only at the very end of the renormalization process. We have introduced in (3.23) the induced metric γ µν = exp(2A)η µν and the trace of the extrinsic curvature K, given by where ζ(z) is defined in (2.21). As shown in [4], using the equations of motion (2.19) the on-shell (o-s) action densities take the form where V 4 is the four dimensional volume. Thus we find the bare energy density The near boundary asymptotic behaviour of the warp factor A(z) was obtained in (3.3).
Using that result in (3.26) one finds terms that diverge when z 0 goes to zero. We consider a minimal subtraction (MS) scheme where the divergent terms are eliminated by adding the appropriate counterterms. After this renormalization process one finally takes the limit z 0 → 0. Most of the non-divergent terms vanish in this limit and the surviving finite piece becomes the renormalized vacuum energy density: where the superscript "ren" means renormalized. The numerical results for this renormalized vacuum energy density will be shown in the next section. Here we just notice that the vacuum energy density is negative and therefore lower than the (zero) energy of the CFT vacuum. We remark, however, that the result (3.27) may change if we use a different renormalization scheme. Note also from (3.9) that φ 0 G =φ 0 Λ 4 has conformal dimension 4, which is indeed the protected conformal dimension of T µν .

Glueball spectrum
Glueballs are bound states of gluons predicted by QCD. So far, they have not been detected although there is a recent claim that the f 0 (1710) scalar particle may actually be the scalar glueball state 0 ++ [31,32]. Furthermore, it was also recently proposed that the odd glueball (oddball) 0 −− could be detected soon by the experiments BESIII, BELLEII, Super-B, PANDA, and LHCb [33], although there is some controversy on this prediction [34]. Other interesting glueball states, as for example, 2 ++ , 0 −+ and 1 −− are under investigation and have candidates in the particle spectrum [35].
We will find the spectrum of scalar 0 ++ and tensor 2 ++ glueball states from solving the differential equations (2.52) and (2.55), respectively. In those Schrödinger like equations the glueball states are represented by wave functions ψ s and ψ t . Under suitable boundary (asymptotic) conditions on these wave functions, the mass spectrum of the respective sector is found. As explained in the previous section, the parameters in our models areφ 0 , Λ and and the strategy is the following: for each value of the parametersφ 0 and Λ will be fixed using as input the lattice QCD results for the first two scalar glueballs [52]. At the end of the numerical calculation, we compare our results for all the other glueball states against lattice QCD data [52], and the results obtained in the improved holographic QCD model [4].
In the numerical calculation it is convenient to rewrite the interpolations (3.19) and (3.20) in terms of a dimensionless coordinate u = Λz: Notice that the parameter Λ has disappeared in (4.1) and (4.2). This is because the u coordinate is dimensionless and the Schrödinger like equation in this coordinate leads to a spectrum where the masses are given in units of Λ. We remind the reader that fixing Λ will fix also the VEV coefficient G = Λ 4− in the UV as well as the IR coefficient C = Λ 2 characterizing confinement.

Analysis of the effective potentials
The spectrum of glueballs will depend on the form of the effective potentials V s and V t , which appear in the Schrödinger like equations (2.52) and (2.55). Here we present an analysis of those potentials.
Let us start with the effective potential V s for the scalar sector, defined in equation (2.53). In terms of the dimensionless variable u, this potential takes the form Our hypothesis is that the conformal dimension is small. Then the leading term of the scalar potential (4.3) takes the form where we have introduced the dilaton mass term M 2 = ( − 4). This term is responsible in the UV for the explicit break of conformal symmetry. Now we turn attention to the effective potential V t for the tensor sector, defined in equation (2.56). In terms of the u coordinate it takes the form The UV asymptotic behaviour of this potential is obtained from the asymptotic behaviour of A(u), given by equation (3.3). The result is simply Notice that the conformal dimension does not affect the UV asymptotic behaviour of the tensor potential.
In the IR regime, at large u, the asymptotic behaviour for the warp factor and dilaton are given by equations (3.11) and (3.18) respectively. Then, X(u) and B s,t (u) have the asymptotic form (4.10) Therefore, the IR asymptotic behaviour of the effective potentials (for both sectors) take the form V s,t (u) Λ 2 = 4u 2 . (4.11) In figure 4 we show the effective potentials, obtained numerically, for models A1 and A2 at = 0.01 andφ 0 = 50. The plots are consistent with the asymptotic results (4.6), (4.9) and (4.11). As expected, the difference between models A1 and A2 lies in the intermediate region.

UV and IR asymptotic solutions for the wave functions
In this section we will find the UV and IR asymptotic solutions for the wave functions ψ s and ψ t in the Schrödinger like equations (2.52) and (2.55).
First we look for the UV asymptotic solution for the wave function ψ s of the scalar sector. Substituting the UV asymptotic form (4.6) of the potential V s in the differential equation (2.52), we find wherem s = m s /Λ parametrizes the 4-d masses of scalar glueballs in units of Λ. Near the boundary the wave function ψ s behaves as a power law, ψ s (u) = u α 1 . Substituting this ansatz into the differential equation (4.12), and solving the resulting indicial equation, we find two solutions for α 1 , namely, α − 1 = − + 5/2 and α + 1 = − 3/2, which means that the asymptotic general solution is of the form The coefficient c 2 is set to zero because we are looking for a normalizable solution.
We perform a similar analysis to obtain the UV asymptotic behaviour of the wave function ψ t of the tensor sector. Using the UV asymptotic form of the potential (4.9), the differential equation (2.55) becomes Finally we look for the IR asymptotic behaviour of ψ s and ψ t . As shown in equation (4.11), the effective potentials of the scalar and tensor sectors have the same asymptotic behaviour in the IR. Then both Schrödinger like equations assume the form − ψ s,t (u) + 4u 2 ψ s,t (u) =m 2 s,t ψ s,t (u). (4.16) Solving this equation, we find the IR asymptotic solutions that converge at infinity can be written as

Glueball spectrum at fixed
The task now is to solve the eigenvalue problem for the differential equations (2.52) and (2.55). We solve this problem numerically using a shooting method, which was implemented in a Mathematica code.
In order to find a unique solution to a second order differential equation, we need two boundary conditions. There are two typical ways of doing this. We can use the asymptotic UV solutions (4.13) and (4.15) and its derivatives, respectively for the scalar and tensor sectors, as boundary conditions at some u = u min , with u min very small. Then, we integrate numerically from small u to large u and require that the wave function at large u should behave as in (4.17). Using those conditions and fixing the parameters andφ 0 , we get a discrete spectrum (in units of Λ). Alternatively, we may take the asymptotic IR solutions (4.17) and its derivatives as initial conditions at some u = u max , with u max very large, and integrate numerically from large u to small u requiring the numerical solutions for ψ s and ψ t at small u to behave as (4.13) and (4.15), respectively.
The numerical results presented in this section were obtained using the first procedure described above. Here we present our results for the models A (introduced in the previous section) as well as for the models B1 and B2, introduced in appendix B).
In this subsection we present the results for the glueball spectrum at a fixed value of the conformal dimension, namely = 0.01. At fixed the parameterφ 0 can be fixed by using as input the ratio between the first two scalar glueballs wherem s,0 andm s,1 represent the first two scalar masses and m 0 ++ and m 0 ++ * are extracted from lattice QCD data [52]. Onceφ 0 is determined from the ratio R 00 , we also fix the parameter Λ by comparing the first scalar mass m s,0 = Λm s,0 with the first glueball state m 0 ++ , extracted from lattice QCD data [52]. Below we describe the results for the glueball spectrum obtained for each one of models A and B.

Models A
Implementing the procedure described above for the model A1, where the dilaton is given by equation (3.19), we find for = 0.01 thatφ 0 = 53.62 and Λ = 737 MeV. Any other parameter is defined in terms of ,φ 0 and Λ and most of them are shown in Table 2.
The results for the spectrum of scalar and tensor glueballs in model A1 are shown in the second column of Table 3. These results are in good agreement with the lattice QCD calculations [52], and also with the IHQCD model [4]. The largest difference between our results and lattice QCD data is about 4.2% in the case of m 0 ++ * * . We remind the reader that the first two masses in Table 3, m 0 ++ and m 0 ++ * , were used to fixφ 0 and Λ. Therefore, the predictions of the present models are the ones displayed from the third state (0 ++ * * ) and below in that table.
The numerical results for the glueball spectrum are well fitted by linear trajectories. For the scalar sector we find the linear trajectory m 2 s,n = Λ 2 (8.65 n + 4.85), n = 0, 1, 2, ..., Modelφ 0  Table 2: The values of the parameters we use to get the spectrum for the glueballs with = 0.01.
while for the tensor sector we obtain m 2 t,n = Λ 2 (8.13 n + 7.92), n = 0, 1, 2, ... The largest difference between the masses obtained with these fits and the lattice QCD results occurs for the state 0 ++ and is of about 10%. The same procedure was done for the model A2, where the dilaton is given by equation The mass spectrum obtained in model A2 is shown in the third column of table 3. We find again a good agreement between our results and those of the lattice QCD [52] and the IHQCD model [4]. In comparison to the lattice QCD masses, the largest difference is of about 2.4% in the case of m 0 ++ * * * .
The spectrum of the model A2 is also well approximated by linear fits. The linear trajectories in this case are m 2 s,n = Λ 2 (8.62 n + 6.37), n = 0, 1, 2, ...  for the tensor sector. When compared to the the lattice QCD results, the maximum error obtained for these fits is of about 16.8%, and it occurs for the state 0 ++ .

Models B
The two models we named B1 and B2 are described in appendix B and correspond to the case where the warp factor A(z) has an analytic form, while the dilaton is solved numerically from the first equation in (2.20).
The parameters obtained for the models B1 and B2 at = 0.01 are shown in the third and fourth row of Table 2. The glueball spectra obtained for the models B1 and B2 are shown in the fourth and fifth column of Table 3. The numerical results for models B1 and B2 are also in good agreement with the Lattice QCD and IHQCD model. The approximate linear trajectories for model B1 are   Table 3: The glueball masses (in MeV) obtained in our model, compared against the results of IHQCD [4] and Lattice QCD [52]. The first two values of masses for 0 ++ and 0 ++ * are used as input data in our procedure. The results here were obtained with = 0.01.
while for model B2 we obtain m 2 s,n = Λ 2 (8.80 n + 5.47), m 2 t,n = Λ 2 (8.17 n + 9.03), n = 0, 1, 2, · · · . (4.24) In figure 5 we plot the results for the spectra of scalar and tensor glueballs obtained in the 4 models considered in this work (A1, A2, B1 and B2) and, for comparison, we include the data of lattice QCD [52]. The results for models A1 and B1 are shown in figure 6. The plots show clearly the pattern m s,n < m t,n , also observed in the IHQCD model [4]. Notice that the difference between the scalar and tensor glueball masses decreases as n increases. This indicates a degeneracy of the scalar and tensor glueballs at very large n.

Running parameters
So far in this section, all the calculations were done for a specific value of the conformal dimension, = 0.01. But what happens when this parameter varies? Here we find the evolution of the parametersφ 0 and Λ with the conformal dimension for the models A1 and A2. As explained previously in this section, for any given we use the masses of the first two scalar glueballs, extracted from lattice QCD, as an input for fixingφ 0 and Λ.
The results forφ 0 are displayed on the left panel of figure 7. A numerical fit shows thatφ 0 diverges as 1/ as the parameter goes to zero. The evolution of Λ with is shown on the right panel of figure 7. The evolution is very slow suggesting that Λ should be approximated by a constant. Linear fits for these results give Λ = 735. We have also obtained numerically the vacuum energy density, given by (3.27). The results are shown in figure 9. One clearly sees that the vacuum energy density converges to a finite value when goes to zero. This can be understood as follows. We showed previously that, when goes to zero, the source φ 0 goes as c 1 + c 2 / . Therefore, the renormalized vacuum energy density (E ren QCD ∝ φ 0 ) goes as c 2 + c 1 . The numerical results shown in figure 9 indicate that the vacuum energy density evolves very slowly with the conformal dimension , in both models A1 and A2, and can be approximated by a constant. Setting

The trace anomaly: from deformed CFTs to QCD
In this section we take advantage of the AdS/CFT dictionary and reproduce the universal result for the trace anomaly of deformed CFTs [9,10]  Here we can make use of the mass parameter Λ, defined previously in (3.9), which has conformal dimension 1. Multiplying and dividing equation (5.2) by Λ , we get In this way we have dressed the inverse 't Hooft coupling 1/λ and the QCD operator Below we will use this map for calculating the gluon condensate and to find a connection between the trace anomaly of deformed CFTs and the QCD trace anomaly.

The gluon condensate
According to the AdS/CFT dictionary the VEV of an operator O is obtained from the variation of the on-shell action [9,10,20,21] where δφ 0 is the variation of the source. In the case of holographic QCD backgrounds, the on-shell action was obtained in subsection 3.4 with the result with small z 0 and, as usual, the limit z 0 → 0 will be taken at the very end of the calculation process. We then variate this on-shell action by considering the warp factor A as a field whose dynamics is completely determined by the dilaton field Φ, A = A(Φ). Therefore, we get where we used the asymptotic form of the dilaton (3.8) to rewrite Φ in terms of the source φ 0 . From equations (5.5) and (5.7) we find the bare VEV of the operator O, Using the field equations (2.20) and the definition (2.24), we may rewrite this VEV as As it was done previously for the case of the bare energy density, we use the UV asymptotic form of A and X to identify the divergent and non-divergent terms in (5.9). Again, we consider a MS scheme for renormalization. Eliminating the divergent terms and taking the z 0 → 0 limit, we obtain the renormalized VEV In figure 10 we plot our numerical results for the gluon condensate 1 4π 2 Tr F 2 ren as a function of for the models A1 and A2. We remark that the gluon condensate in (5.11) was obtained in a particular renormalization scheme. Setting M 3 N 2 c to unity and taking the limit → 0, the results for the gluon condensate 1 4π 2 Tr F 2 ren are 0.063 GeV 4 and 0.047 GeV 4 for the models A1 and A2, respectively. For comparison, we present other results in the literature. The values obtained using SVZ sum rules are 0.013 GeV 4 [56] and 0.012 GeV 4 [57]. Previous results in holographic QCD include 0.043 GeV 4 [2] and 0.01 GeV 4 [58]. We also mention two different results in SU(3) lattice gauge theory: 0.10 GeV 4 [59] and 0.04 GeV 4 [60].

The trace anomaly
The renormalized vacuum energy density E ren QCD for our model is given by equation (3.27). At zero temperature, the pressure is just −E ren QCD , so the trace of the energy mo-  Figure 10: Evolution of the renormalized gluon condensate 1 4π 2 Tr F 2 ren with the conformal dimension for models A1 and A2. mentum tensor is given by From equations (5.10) and (5.12), we find the relation This is the universal trace anomaly of 4-d CFTs deformed by an operator O with dimension ∆ = 4 − and coupling φ 0 [9,10]. The quantity − φ 0 is the classical β function associated with the coupling φ 0 . We have reproduced this trace anomaly within the context of effective holographic QCD backgrounds, where the dilaton and warp factor depend solely on the radial coordinate z. This trace anomaly describes the explicit breaking of conformal symmetry and, as described in the previous section, a nontrivial consequence of this symmetry breaking is the discrete spectrum of scalar and tensor glueballs. It is interesting to note that the limit → 0, with φ 0 fixed, corresponds to the case where conformal symmetry is spontaneously broken. In that case, as explained in appendix A, the first scalar glueball becomes a Nambu-Goldstone boson.
The trace anomaly (5.13) holds for more general backgrounds where the dilaton and metric are more involved. In any case the conformal dimension of O always maps to a mass term for the dilaton via the relation M 2 = ∆(∆ − 4). Although a general proof of (5.13) was developed in [9,10], it is always illuminating to reproduce this trace anomaly case by case. In particular, it would be interesting to prove (5.13) for the case of black hole solutions, such as those considered in [6,7].
We finish this section proposing a dictionary between the conformal trace anomaly (5.13)  suggesting the identification = −β/λ. We remind the reader that the CFT deformation takes place at some UV energy scale E * so the coupling λ appearing in equation (5.14) is actually evaluated at that scale. A few remarks are in order here. The dictionary proposed in this work differs from the original proposal [6,7] because we map the conformal dimension to the β function of the 4-d theory, instead of the anomalous dimension of Tr F 2 . Our dictionary also differs significantly from [3,4] where the 4-d β function is mapped to the 5-d field X. In [3,4] the 4-d energy scale and coupling are mapped to the 5-d warp factor and dilaton in a very natural way. However, evaluating correlation functions at any RG energy scale E * becomes a difficult task due to the necessity of introducing a geometric cutoff z * . 7 In the effective holographic approach considered here, a RG energy scale E * does not imply cutting the 5-d geometry so that one can make full use of the AdS/CFT dictionary by embedding holographic QCD in the framework of holographic deformed CFTs, developed in [8][9][10].  Figure 11 shows the numerical results for the 4-d β function in terms of λ. According to the dictionary proposed in this work, these quantities are identified with − /φ 0 and 1/φ 0 , respectively. We made a numerical fit for small λ and interestingly the fit takes the form β = −b 0 λ 2 − b 1 λ 3 . This is the same form arising in large-N c perturbative QCD at two-loops, with b 0 and b 1 being scheme independent. For the model A1 we find b 0 ≈ 0. We finish this section describing the relation between the anomalous dimension an of Tr F 2 and the conformal dimension . Following Ref. [7], the anomalous dimension an can be extracted from the QCD trace anomaly (5.15) with the result an = −λ 2 ∂ λ βλ −2 . (5.16) At small λ the anomalous dimension (5.16) is approximated by an ≈ b 1 λ 2 . Using the identifications proposed in this work, namelyφ 0 = 1 λ and = − β λ , this relation becomes where in the last equality we have introduced the beta function associated withφ 0 , defined as the derivative ofφ 0 with respect to log E * . Interestingly, the relation (5.17) looks very similar to the anomalous dimensions that arise in holographic RG flows [61].

Conclusions
In this work we have investigated effective holographic models where QCD is described in terms of a 4-d CFT deformation. The deformation is of the form δL = φ 0 O, where O is a relevant operator and φ 0 the coupling, and takes place at a UV energy scale E * . It is characterized by the conformal dimension of the relevant operator ∆ = 4 − , which according to the AdS/CFT dictionary, maps to the 5-d mass of the dual dilaton field. The IR dilaton asymptotics was constrained by the criteria of confinement and linear glueball spectrum; namely a dilaton quadratic in the radial coordinate z. We have proposed UV/IR semi-analytic interpolations that lead to a spectrum of scalar and tensor glueballs consistent with lattice QCD. A key ingredient in our description was the evolution of the model parameters with . In particular, the evolution of the coupling φ 0 with was essential to guarantee an explicit breaking of conformal symmetry consistent with the glueball spectrum.
Making use of the AdS/CFT correspondence we have evaluated the renormalized vacuum energy density T 00 and the VEV of the relevant operator O in the 4-d theory. Both quantities are different from zero as a consequence of the CFT deformation. We have mapped those quantities to the QCD vacuum energy and gluon condensate respectively. We have also reproduced the universal result for the trace anomaly in 4-d deformed CFTs, namely T µ µ = − φ 0 O , and reinterpreted this result in terms of the QCD trace anomaly. This led us to suggest a map between the conformal dimension and the β function of the QCD-like theory. The dictionary found in this work differs significantly from the one proposed in [3,4], but establish a novel connection between QCD and deformed CFTs. Moreover, from the evolution of φ 0 with and the dictionary proposed in this work we found a 4-d β function that behaves qualitatively as the large-N c QCD β function up to the perturbative regime. This nontrivial result indicates that the holographic description of QCD as a CFT deformation can be consistent with asymptotic freedom without the necessity of building a specific potential.
There are some pieces of the dictionary that remain to be found, such as the relation between the 5-d warp factor and the 4-d energy. A map between the backgrounds developed in this work and the backgrounds developed in [3,4] would also be desirable. That map should be such that the metric and dilaton near a geometric cutoff z * in the background of [3,4] becomes an AdS metric slightly deformed by a massive dilaton. The holographic description of the Callan-Symanzik equations, following [29,61], would also shed some light in the connection between holographic QCD and deformed CFTs.
We finish this work mentioning some of its possible extensions. The description of mesons and chiral symmetry breaking in terms of gauge fields and a tachyonic field can be done, inspired by the progress made in [5,62,63]. Investigating black hole solutions at finite temperature will allow the description of a non-conformal plasma, complementing the progress made in [6,7] and [30]. In this context, a particular interesting phenomena is the so-called glueball melting [64]. Finally, the study of higher spin glueballs and the pomeron could be pursued, along the lines of [65] and more recently [50,[66][67][68].
Since we are looking for the wave function of the massless mode, we substitutem 2 s = 0 in equation (A.1) to get ∂ z + B s (z) −∂ z + B s (z) ψ s (z) = 0. (A. 3) The solutions of this equation may be written as [24] ψ (1) s (z) = e Bs(z) , ψ must be finite. Following the procedure developed in section 4.1, we write down the leading terms of the asymptotic expansions for B s (z) in the UV and IR regimes: B s (z) = −C z 2 + · · · , z → ∞; (A. 6) where φ 0 , G and C are the constant parameters previously introduced, with C being a positive real parameter. We start by analyzing the first solution, ψ where Erf[ √ C z] is the error function. Therefore, the first solution is normalizable in the IR regime.
(i) ∆ − = 0 and ∆ + = 4. This represents the extremal case, for which M = 0. In this case the conformal dimension does not show in, but there are still the source and operators, was developed in [15].
• ∆ − = 3/2 and ∆ + = 5/2. For these specific values of ∆ ± , the integral (A.5) becomes 14) The last result depends on the source value (which is finite) and therefore the wave function is normalizable. So, there is a massless mode also for these specific values of ∆ ± .
As the integral is finite in case iv, it means that both terms of the wave function (A.8) are normalizable, and the source and the condensate can exchange their roles [16]. Now let us look at the second solution, ψ This solution might be or not normalizable in the UV. But as the solution in the IR is always non-normalizable, there is not a massless mode in this case.
In conclusion, the normalizability of the first solution ψ (1) s (z) depends on the values of ∆ − and φ 0 . As this solution is normalizable in the IR, by choosing appropriately these two parameters, it can be made normalizable also in the UV regime, and then a massless mode can be obtained in this case. On the other hand, the second solution is non-normalizable in the IR. This result excludes the possibility to get a massless mode from the second solution.

B. Models B: analytic form for the warp factor
Alternatively to what was presented in the main body of this work, we can choose to interpolate the warp factor between the UV and IR, instead of the dilaton. For future reference, as instance to explore the finite temperature effects, it is convenient to have an analytic warp factor. We call this approach of interpolating the warp factor as model B. Inspired by the interpolations done above for the dilaton field, we carefully seek the form of the interpolating functions so that they produce the asymptotic forms of equations (3.3) and (3.11), respectively.
Our first choice, motivated by the model A1, is where the functions a 1 (z) = 7(Λz) 5 /50 and b 1 (z) = 5(Λz) 4 + 3(Λz) 6−2 /2 were introduced to guarantee the compatibility of the results of the model with those of the lattice QCD [52]. These functions also guarantee that the warp factor decreases monotonically from the UV to the IR, as required in Dilaton-Gravity models [24]. We named this interpolation form as model B1.
To complete the analysis of these models, we show in figure 12 the potentials of the Schrödinger like equations for both sectors, scalar and tensor, as indicated. The spectra calculated using these models are presented in table 3.