Published for SISSA by Springer Received: June 24, 2019 Revised: September 19, 2019 Accepted: October 8, 2019 Published: October 18, 2019 JHEP10(2019)202 Holographic conformal transition and light scalars Alex Pomarol,a,b Oriol Pujolasa and Lindber Salasa a IFAE and BIST, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona b Departament de Fı́sica, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona E-mail: alex.pomarol@uab.cat, oriol.pujolas@gmail.com, lsalas@ifae.es Abstract: We present an holographic approach to strongly-coupled theories close to the conformal to non-conformal transition, trying to understand the presence of light scalars as recent lattice simulations seem to suggest. We find that the dilaton is always the lightest resonance, although not parametrically lighter than the others. We provide a simple analytic formula for the dilaton mass that allows us to understand this behavior. The pattern of the meson mass spectrum, as we get close to the conformal transition, is found to be quite similar to that in lattice simulations. We provide further predictions from holography that can be checked in the future. These five-dimensional models can also implement new solutions to the hierarchy problem, having implications for searches at the LHC and cosmology. Keywords: Beyond Standard Model, Conformal Field Theory, Large Extra Dimensions, Technicolor and Composite Models ArXiv ePrint: 1905.02653 Open Access, c The Authors. https://doi.org/10.1007/JHEP10(2019)202 Article funded by SCOAP3 . Contents 1 Introduction 1 2 Conformal transition by fixed-point merging 3 3 A five-dimensional model for the conformal transition 5 3.1 The large Nc and NF power counting 7 3.2 The tachyon solution 8 JHEP10(2019)202 3.2.1 Region m̂2b > −2 9 3.2.2 Region m̂2b < −2 11 3.3 Radion/dilaton stabilization 12 3.4 Excitations around the 5D tachyon 14 3.4.1 The singlet scalar sector and light dilaton 14 3.4.2 Non-singlet scalars, vector and axial-vector excitations 17 4 Comparison with lattice QCD in the large NF 20 5 Models for the hierarchy problem 22 6 Conclusions 23 A Scalar and gravity coupled equations of motion 24 A.1 Scalar-metric system 24 A.2 Singlet scalar-dilaton system 25 A.2.1 Light dilaton limit 26 B A tale of two scalars: the 4D effective potential of a tachyon and a dilaton 29 B.1 Effective quartic coupling for the tachyon 31 1 Introduction Theories close to being conformally invariant are of utmost interest as they can generate large hierarchies of scales that can be useful in particle physics and cosmology. This moti- vates the understanding of how theories behave at the critical point at which, by varying the parameters of the theory, we pass from a conformal regime to a non-conformal one. This is especially interesting in strongly-coupled theories as they can give rise to non- trivial dynamics. An example is QCD where by increasing the number of flavors NF , the theory is expected to become conformally invariant at some critical value NF = NFcrit . It is unclear where this exactly happens, but lattice simulations suggest that this could be around NFcrit ∼ 10 for Nc = 3 where Nc is the number of colors. For NF ≥ NFcrit , –1– QCD becomes a conformal field theory (CFT) till reaching NF = 11 2 Nc , at which the theory reaches the Banks-Zaks fixed point, becoming IR free for NF > 11 2 Nc . The region crit 11 NF ≤ NF ≤ 2 Nc is called the conformal window. Recent lattice simulations suggest that, contrary to real QCD, theories close to the conformal transition have as the lightest resonance a 0++ state (apart, of course, from the Goldstone bosons, the pions) [1–5]. It is unclear what is the origin of the lightness of this state. Some arguments suggest that this could be a dilaton, the Goldstone associated to the spontaneous breaking of scale invariance. If this is the case, it would be interesting to know whether in the large-Nc limit, where NFcrit /Nc ≡ xcrit becomes a continuous parameter, the dilaton mass tends to zero as we approach the critical point from below NF /Nc → xcrit . JHEP10(2019)202 In this article we would like to analyze the physics of conformal transitions using holography. We will follow ref. [6] that argued that the exit of the conformal window of large-Nc QCD occurs when the IR fixed point disappears by merging with a UV fixed point. Close to the conformal edge the theory contains a marginal operator Og whose dimension gets a small imaginary part when conformal invariance is lost (see next section for details). Assuming that this is the case, the AdS/CFT correspondence [7–9] can provide a simple realization of this idea as a complex operator dimension matches to a scalar having a mass below the Breitenlohner-Freedman (BF) bound MΦ2 = −4/L2 . When this happens, the scalar becomes tachyonic and gets a non-zero profile that results into a departure from the Anti-de-Sitter (AdS) geometry [6]. The presence of a marginal operator Og in the model could suggest the presence of a light dilaton, along the lines of refs. [10–14]. The argument goes as follows. The dilaton potential can be written as Veff (φd ) = λeff (φd ) φ4d , (1.1) such that, when a minimum exists, leads to a dilaton mass given by m2φd = βλeff (4 + βλ0 eff ) , (1.2) hφd i2 where βλeff = dλeff /d ln φd and βλ0 eff = dβλeff /dλeff . A nonzero βλeff arises only from an explicit breaking of scale invariance. When this latter comes only from 1 g Og ∈ L, we expect βλeff ∝ βg , and eq. (1.2) predicts m2φd ∝ βg . Therefore, a dilaton can be parametrically light if the dimension of Og is given by 4 + δ with δ  1 (i.e., βg  1) being a controllable small parameter till the end of the RG-flow. The holographic implementation of this is the Goldberger-Wise mechanism [15], where the operator Og matches to an almost massless scalar in 5D (protected by a shift symmetry) [10–14]. Nevertheless, we will see that this is not the case at the conformal transition, as the marginal operator Og corresponds to a double-trace operator whose dimension is not protected along the RG-flow. Having the explicit breaking of conformal invariance arising from an almost marginal operator however will have as a consequence that the dilaton is light, although not parametrically light. We will be working with a simple weakly-coupled AdS5 theory, with the extra- dimension cut off by an IR-brane, that will contain the basic ingredients to describe the 1 We remark that g is a coupling of the theory not necessarily related with the gauge coupling. –2– conformal transition. We will calculate the mass spectrum of resonances and show that the lightest resonance is the dilaton (the radion of the compact extra-dimension). We will present a simple analytical formula for the mass of the dilaton that will allow to under- stand its lightness as a function of the change of the tachyon as we move the IR-brane. This will show that either at small or large positions of the IR-brane, the dilaton is always parametrically light. In between these two regions, we will see that the dilaton mass does not have “room” to grow and as a consequence the dilaton is always kept light. We will compare our results with lattice simulations, showing good agreement in the pattern of masses when the conformal critical point is approached. Furthermore, we will JHEP10(2019)202 provide further predictions to be checked in the future by lattice simulations. The 5D model presented here could also be useful to generate small scales and explain, for example, the difference between the electroweak scale and the Planck scale. More- over, the presence of a light scalar can have an important impact in the searches for new resonances at the LHC as predicted in composite Higgs models. There have been previous approaches using holography to understand the conformal transition and the existence of a light dilaton [16–29]. We find however that these studies were not exhaustive nor conclusive. Our goal is not only to provide evidence for a relatively light dilaton, but also to explain the reasons behind this. The article is organized as follows. In section 2 we introduce the idea of leaving the conformal window by fixed-point merging and remark its implications. In section 3 we present the five-dimensional model and its relation with the large Nc and NF expansion. We also discuss the tachyon solution and the stabilization of the radion. Next we present the predictions for the resonance mass spectrum, presenting an analytical formula for the case when the dilaton is light, as well as discussing the other scalar and vector resonance masses. In section 4 we compare the mass spectrum calculated within our model with that obtained from lattice simulations, and in section 5 we discuss how these models could also be useful for explaining the smallness of the electroweak scale. Conclusions are given in section 6. We also present two appendices. In appendix A we give the coupled system of equations of motion for the scalar and gravitational sectors, and derive the approximate analytical formula for the dilaton mass. In appendix B we present the 4D effective theory of a tachyon and dilaton valid when they are the lightest states. 2 Conformal transition by fixed-point merging There are several ways to lose an IR fixed point as we move the parameters of the theory. Either the fixed point goes to zero, to infinity or it merges with a UV fixed point. Following ref. [6] we will consider conformal transitions characterized by the third case, the merging of the IR fixed point with a UV fixed point, as depicted in figure 1. In this case, the beta function can be written as βg ' − − (g − g∗ )2 , (2.1) where g is a coupling of the theory (not necessarily related to the gauge coupling in gauge theories), and  depends on the parameters of the theory, e.g., NF . The IR and UV fixed –3– βg ϵ <0 < g* < | | ϵ =0 | g < > ϵ >0 < > < > JHEP10(2019)202 Figure 1. Beta function of the coupling g for different values of . For  = 0, the IR and UV fixed points merge at g∗ . point are respectively at √ g = g∗ ∓ − . (2.2) As we vary  from negative to positive values, we have the merging of the IR and UV fixed points at  = 0, while for  > 0 the theory abandons conformality, i.e., the IR fixed point is at complex coupling. As argued in ref. [30], for  negative and close to zero, the operator Og with coupling g must have dimension dβg √ Dim[Og ] = 4 + ' 4 + 2 − , (2.3) dg and can be considered to be responsible for the RG flow towards the IR fixed point. For  = 0 we have that Og becomes marginal, and develops a complex dimension for  > 0, signaling the end of conformality. The above properties of this conformal transition have a straightforward holographic interpretation using the correspondence (or duality) between strongly-coupled CFT 4 (in the large Nc and large ’t Hooft coupling) and weakly-coupled five-dimensional Anti-de- Sitter theories (AdS5 ) [7–9]. Operators in the CFT4 (O) correspond to scalars in the AdS5 (Φ) where dimensions and masses are related via the AdS/CFT relation [7–9]: q Dim[O] = 2 + 4 + MΦ2 L2 . (2.4) Eq. (2.4) tells us that in order to have a dual of a CFT operator with complex dimension, the AdS5 must have a scalar slightly below the BF-bound, MΦ2 = −(4 + )/L2 . Eq. (2.4) also tells us that this operator of complex dimension (O∗ ) has, in the limit  → 0, dimension 2 instead of 4. Therefore the natural identification for the Og operator discussed above is Og = |O∗ |2 , since in the large Nc this implies Dim[Og ] = 2Dim[O∗ ] and that gives us eq. (2.3). In other words, Og must be a double-trace operator.2 The existence of O∗ in the conformal transition is only an implication from large-Nc theories, and could be not true in general. In QCD, as argued in ref. [6], O∗ is expected to be 2 It has been proven in refs. [30, 31] that this is always the case for theories in the large-Nc limit. –4– the q q̄ operator whose dimension will go from ∼ 3 when entering the conformal window (at the Banks-Zaks fixed point) to 2 when leaving it at the other side when it becomes complex. When the theory is close but outside the conformal window (i.e. 0 <   1), one can calculate the RG flow “time” required to cross the region where g ∼ g∗ and |βg |  1. This gives us the IR-scale ΛIR at which the theory is expected to confine as g becomes large. From eq. (2.1) one gets √ ΛIR ∼ e−π/  ΛUV , (2.5) where ΛUV is roughly the scale at which g . g∗ . Eq. (2.5) is usually referred as walking or Miransky scaling. JHEP10(2019)202 3 A five-dimensional model for the conformal transition We want to study the conformal transition described above using holography. For this reason we will consider the simplest but at the same time most generic five-dimensional model containing the basic ingredients needed to describe the conformal transition via fixed-point merging. Our purpose is to generically understand the mass spectrum at the conformal transition and the presence or not of light scalars. Let us recapitulate the basic ingredients of the theory in the 4D side. This is a strongly- j coupled deformed CFT with a scalar operator, qLi q̄R (i, j = 1, . . . , NF ) for concreteness, √ j whose dimension is 2 + − with 0 <   1. This means that the scalar qLi q̄R gets a vacuum expectation value (VEV), signaling the loss of conformality. The global symmetry of this theory is SU(NF )L ⊗ SU(NF )R ⊗ U(1)B that is broken by the VEV of the scalar j hqLi q̄R i ∝ 1l down to the diagonal subgroup SU(NF )L ⊗ SU(NF )R → SU(NF )V .3 The corresponding holographic model will consist in a SU(NF )L ⊗ SU(NF )R ⊗ U(1)B gauge theory in 5D with a complex scalar Φ transforming as a (NF ,NF )0 . This scalar plays the role of the q q̄ operator whose VEV is responsible for the breaking of the conformal and gauge symmetry, and therefore its mass will be related to the dimension of the q q̄ operator. We also impose parity, defined as the interchange L ↔ R. The action is given by   √ Z Z 4 1 S5 = d x dz g M5 2 (R + Λ5 ) + L5 , (3.1) κ where, up to dimension-four operators,4 the most general Lagrangian is given by 1   1 1 L5 = − Tr LM N LM N + RM N RM N − BM N B M N + Tr |DM Φ|2 − VΦ (Φ) , (3.2) 4 4 2 with LM N , RM N and BM N being the field-strength of the SU(NF )L , SU(NF )R and U(1)B gauge bosons respectively, and the indices run over the five dimensions, M = {µ, 5}. We parametrize the fields as Φ = Φs + Ta Φa with Tr[Ta Tb ] = δab . The fields Φs and Φa will 3 The U(1)A is anomalous and will not be considered here. 4 Following the Effective Field Theory (EFT) approach, higher-dimensional operators are supposed to be suppressed by the cutoff scale of the model (Λcutoff ) estimated to be the scale at which the 5D theory becomes strongly coupled (i.e., when loops are as important as tree-level contributions), that is Λ cutoff ∼ 24π 3 M5 [32] — see also section 3.1. –5– respectively transform as singlet and adjoint under the SU(NF )V . The covariant derivative is defined as DM Φ = ∂M Φ + ig5 LM Φ − ig5 ΦRM , (3.3) and the potential is given by5 1 1 1 VΦ (Φ) = MΦ2 Tr |Φ|2 + λ1 Tr |Φ|4 + λ2 (Tr |Φ|2 )2 . (3.4) 2 4 4 The 5D metric in conformal coordinates is defined as ds2 = a(z)2 ηµν dxµ dxν − dz 2 , JHEP10(2019)202  (3.5) where ηµν = diag(1, −1, −1, −1) and a(z) is the warp factor. Before the scalar Φ turns on, the presence of Λ5 leads to an AdS5 geometry: L a(z) = , (3.6) z where L2 = 12/Λ5 is the squared AdS curvature radius. As explained above, our important ingredient here is to consider that the conformal breaking arises when Dim[q q̄] becomes imaginary. In AdS this corresponds from eq. (2.4) to take the 5D mass of Φ below the BF bound. For this purpose, we will consider 4+ MΦ2 = − . (3.7) L2 When  > 0 the mass of Φ is below the BF bound and Φ turns on in the 5D bulk, breaking the conformal and chiral symmetry SU(NF )L ⊗ SU(NF )R → SU(NF )V . Φ will grow as ∼ z 2 , as expected from a dimension-two perturbation in the dual 4D theory. When the energy-momentum tensor induced by the nonzero Φ profile gets of order of the inverse 5D Newton constant, κ2 , the backreaction on the metric will be important, starting to depart then from AdS, and signaling the breaking of the conformal symmetry. This simple model, however, does not lead to a mass gap for all bulk fields, as the extra dimension is not ending at any z. In fact, as we will see, the tachyon Φ would stabilize at the minimum of the potential eq. (3.4) and the metric would become again AdS. As we know that in strongly-coupled models outside the conformal window, like QCD, all resonances are heavy, we need to implement the same in our holographic version. The simplest way is to cut off the 5D space by an IR-brane at some point z = zIR to be determined dynamically. The presence of the IR-brane add extra parameters to the theory as Φ might also have a potential on the IR-boundary. We will limit ourselves to up to quadratic terms in Φ: Λ4 1 2 LIR = −a4 Ṽb (Φ) zIR , Ṽb (Φ) = + mb Tr |Φ|2 , (3.8) κ2 2 and study their impact on the properties of the model. We could also add to eq. (3.8) quartic terms but these are not expected to change significantly our predictions, since they 5 We notice that one can absorb one coupling into M5 , as we will do later. –6– are suppressed by 1/(M5 L) with respect to the bulk terms.6 As it is usual in AdS/CFT, we will be regularizing the UV-divergencies by placing a UV-boundary at z = zUV and taking the limit zUV → 0 at the end of the calculation of physical quantities. In this 5D model the two phases are determined, as in the strongly-coupled model described in section 2, by the  parameter: • For  < 0, we have Φ = 0 and zIR = ∞: AdS5 (CFT4 ) phase. • For  > 0, we have Φ 6= 0 and zIR 6= ∞: non-AdS5 (non-CFT4 ) phase. JHEP10(2019)202 3.1 The large Nc and NF power counting By the AdS/CFT correspondence, the 5D scalar and gauge bosons are associated to the me- son operators q̄q and q̄γµ q respectively. Therefore 5D couplings from single-trace operators must scale in this sector as 1/Nc . This can be implemented by assuming 1 16π 2 ∼ , λ1 ∼ g5 ∼ Nc0 . (3.9) M5 L Nc On the other hand, double-trace operators are suppressed with respect to single-trace ones: 1 λ2 ∼ . (3.10) Nc For this reason these latter terms were neglected in previous holographic approaches to QCD [33–35]. Nevertheless, in some physical quantities the parameter λ2 is accompanied by a factor NF , as we will see explicitly below (e.g. eq. (3.14)), and then its effect is not suppressed for large values of NF . Therefore it is important to keep double-trace operators in eq. (3.2) when comparing our results to strongly-coupled theories in the large Nc and NF limit. In particular, λ2 will be responsible for generating a mass splitting in the scalar sector between the singlet (Φs ) and the adjoint states (Φa ), as observed in lattice results with large NF [1–5]. Using the scaling eq. (3.9), one realizes that in the limit NF ∼ Nc , 5D models are in danger of not providing reliable calculations by the usual perturbative expansion. Indeed, in this limit loops of vector or scalar resonances in the adjoint of SU(NF )V contribute 1 NF NF as 16π 2 M L ∼ Nc ∼ 1, meaning that loops could be as important as tree-level contribu- 5 tions. The NF enhancement of these corrections is due to the large number of fields (for example,Φa consists of NF2 − 1 fields). Still, by extending the results to NF /Nc ∼ 1, one hopes to capture well the qualitative dynamics. A case that is more under control arises if one can restrict only to the flavor-singlet sector of the theory [36], which is justified when the other sectors are heavier. In this case the NF dependence enters only into the couplings of the flavor-singlet sector, which can be treated perturbatively (loops are always small since the number of fields is not large) even in the strict Veneziano limit where NF is of order or even larger than Nc . In fact, ref. [36] 6 Even though the same is true for the two terms in eq. (3.8), these are respectively quartically and quadratically UV sensitive so they can be sizable. –7– provides an example of a weakly-coupled closed string dual description of the flavor-singlet sector of a gauge theory in the Veneziano limit. From the AdS/CFT dictionary, we are also able to relate the gravitational sector of the 5D theory with the glueball sector of the 4D CFT, and derive the scaling of the 5D Newton constant with the number of colors: κ2 /(M5 L3 ) ∼ 16π 2 /Nc2 that using eq. (3.9) implies κ2 1 2 ∼ . (3.11) L Nc From the above we can estimate the mixing between the flavor-singlet meson sector and the glueball sector (dual respectively to the scalar and gravitational sectors in 5D) to go as JHEP10(2019)202 κ2 N F NF κ̂2 ≡ ∼ , (3.12) L2 Nc that becomes order one for NF ∼ Nc . Therefore, contrary to previous holographic models, the impact of the gravitational sector in the singlet scalar sector cannot be neglected in this case. 3.2 The tachyon solution The non-zero profile for Φ will be taken to be along the φ = |Φs | direction, whose 5D Lagrangian is given by   1 L φ = NF (∂M φ) − V (φ) + g52 φ2 Tr A2M , 2 (3.13) 2 where 1 1 V (φ) = MΦ2 φ2 + λφ4 , λ ≡ λ1 + N F λ 2 , (3.14) 2 4 √ being AM = (LM − RM )/ 2 the axial-vector gauge bosons that will get masses from their coupling to φ. The IR-brane potential can be written as Λ4 1 LIR = −a4 NF Vb (φ) zIR , Vb (φ) = 2 2 + m2b φ2 . (3.15) κ̂ L 2 Notice that the presence of a factor NF in front the Lagrangian means that the couplings of φ are suppressed by an extra 1/NF with respect to those in the non-singlet sector, as expected in strongly-coupled theories in the large Nc ∼ NF limit. The equation of motion (EOM) for φ from eq. (3.13) must be solved including the metric back-reaction that via the Einstein equations (see appendix A) determines the warp factor: s ȧ 1 κ̂2 L2  φ̇2  − 2 = + − V (φ) , (3.16) a L2 12 2a2 where from now on we will be using the dot notation: φ̇ ≡ ∂z φ. It is important to notice p that by the field redefinition φ → φ/ |λ| we could factorize |λ| in front of the first term of eq. (3.13) and make the EOM that determines the solution for φ independent of |λ| (only sensitive to its sign). This redefinition introduces |λ| in the interactions of φ with –8– the gauge and gravitational fields (second term of eq. (3.13) and eq. (3.16) respectively). Nevertheless, this can be reabsorbed respectively in g52 and κ̂2 , making the solutions and full mass spectrum of the model insensitive to |λ|. Therefore, with no loss of generality, we will consider λ = ±1. We are interested to study the model close to the conformal transition. Therefore we will work in the limit  → 0. The solution for φ then only depends on zIR , κ̂2 and m2b (and the sign of λ). At the UV-boundary we will impose φ = 0, otherwise we would be breaking the chiral symmetry from UV-physics (as adding an explicit mass term to the quarks in the dual theory).7 On the other hand, at the IR-brane we must impose the boundary condition determined by the model: JHEP10(2019)202   M5 0 φ̇ + Vb = 0, (3.17) a zIR where we defined Vb0 ≡ ∂φ Vb . For the metric we must impose the junction condition [37]:   6M5 ȧ − 2 2 2 + Vb = 0. (3.18) κ̂ L a zIR 3.2.1 Region m̂2b > −2 We will start looking for solutions of the tachyon for m̂2b > −2, where we define m2b L m̂2b ≡ . (3.19) M5 In this case non-trivial solutions from eq. (3.13) fulfilling eq. (3.17) are only found if the c . It is easy to find z c , as this corresponds IR-brane is beyond some critical value, zIR > zIR IR to the critical value at which we pass from having all Kaluza-Klein (KK) states of φ with positive squared masses to having 4D tachyons in the theory. Therefore at zIR = zIR c there must be a 4D massless mode, φt (x). The wave-function of this massless mode must satisfy the linearized bulk EOM with p2 = 0. We obtain [16, 17] √   φt (x) 2 z φ(x, z) = z sin  ln , (3.20) N zUV with N a normalization constant, and where the IR-boundary condition eq. (3.17) at c leads to zIR = zIR c  √ c √ √  zIR  zIR tan  ln =− ⇒  ln ' nπ , n = 1, 2, . . . . (3.21) zUV 2 + m̂2b zUV Notice that to have non-trivial solutions, the limit  → 0 must be taken with zUV → 0, such that the angle in eq. (3.20) is kept fixed. The presence of n solutions in eq. (3.21) is a well-known feature of these configurations, and it is associated to the existence of Efimov states. We will be considering n = 1, that will give us the global minimum, being the other possibilities just local minima. Eq. (3.21) reproduces eq. (2.5) for ΛIR ∼ 1/zIR c and 7 Imposing a different boundary condition, such as z φ̇|zUV ∝ φ|zUV , would lead to the same predictions in the limit  → 0 (zUV → 0). –9– 0.6 3.5 0.5 3.0 0.4 2.5 AAAB7nicbZDLSgMxFIZP6q3WW9Wlm2AR6qbMiKDLohuXFewF2qFk0kwbmsmEJCPUoQ/hxoUibn0ed76NaTsLbf0h8PGfc8g5f6gEN9bzvlFhbX1jc6u4XdrZ3ds/KB8etUySasqaNBGJ7oTEMMEla1puBesozUgcCtYOx7ezevuRacMT+WAnigUxGUoecUqss9o9NeLVp/N+ueLVvLnwKvg5VCBXo1/+6g0SmsZMWiqIMV3fUzbIiLacCjYt9VLDFKFjMmRdh5LEzATZfN0pPnPOAEeJdk9aPHd/T2QkNmYSh64zJnZklmsz879aN7XRdZBxqVLLJF18FKUC2wTPbscDrhm1YuKAUM3drpiOiCbUuoRKLgR/+eRVaF3UfMf3l5X6TR5HEU7gFKrgwxXU4Q4a0AQKY3iGV3hDCr2gd/SxaC2gfOYY/gh9/gC7qo8o (z) 2.0 0.3 0.2 AAAB7nicbZDLSgMxFIZP6q3WW9Wlm2AR6qbMiKDLohuXFewF2qFk0kwbmsmEJCPUoQ/hxoUibn0ed76NaTsLbf0h8PGfc8g5f6gEN9bzvlFhbX1jc6u4XdrZ3ds/KB8etUySasqaNBGJ7oTEMMEla1puBesozUgcCtYOx7ezevuRacMT+WAnigUxGUoecUqss9o9NeLVp/N+ueLVvLnwKvg5VCBXo1/+6g0SmsZMWiqIMV3fUzbIiLacCjYt9VLDFKFjMmRdh5LEzATZfN0pPnPOAEeJdk9aPHd/T2QkNmYSh64zJnZklmsz879aN7XRdZBxqVLLJF18FKUC2wTPbscDrhm1YuKAUM3drpiOiCbUuoRKLgR/+eRVaF3UfMf3l5X6TR5HEU7gFKrgwxXU4Q4a0AQKY3iGV3hDCr2gd/SxaC2gfOYY/gh9/gC7qo8o (z) 1.5 1.0 0.1 0.5 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0 5 10 15 20 c c z/zIR AAAB+3icbZC7TsMwFIZPuJZyC2VksaiQmEqCkGCsYIGtIHqR2hA5rtNatZPIdhBtlFdhYQAhVl6EjbfBvQzQ8kuWPv3nHJ3jP0g4U9pxvq2l5ZXVtfXCRnFza3tn194rNVScSkLrJOaxbAVYUc4iWtdMc9pKJMUi4LQZDK7G9eYjlYrF0b0eJtQTuBexkBGsjeXbpWyUn2QjP+tIgW7u8geS+3bZqTgToUVwZ1CGmWq+/dXpxiQVNNKEY6XarpNoL8NSM8JpXuykiiaYDHCPtg1GWFDlZZPbc3RknC4KY2lepNHE/T2RYaHUUASmU2DdV/O1sflfrZ3q8MLLWJSkmkZkuihMOdIxGgeBukxSovnQACaSmVsR6WOJiTZxFU0I7vyXF6FxWnEN356Vq5ezOApwAIdwDC6cQxWuoQZ1IPAEz/AKb1ZuvVjv1se0dcmazezDH1mfPynTlIA= z/zIR AAAB+3icbZC7TsMwFIZPuJZyC2VksaiQmEqCkGCsYIGtIHqR2hA5rtNatZPIdhBtlFdhYQAhVl6EjbfBvQzQ8kuWPv3nHJ3jP0g4U9pxvq2l5ZXVtfXCRnFza3tn194rNVScSkLrJOaxbAVYUc4iWtdMc9pKJMUi4LQZDK7G9eYjlYrF0b0eJtQTuBexkBGsjeXbpWyUn2QjP+tIgW7u8geS+3bZqTgToUVwZ1CGmWq+/dXpxiQVNNKEY6XarpNoL8NSM8JpXuykiiaYDHCPtg1GWFDlZZPbc3RknC4KY2lepNHE/T2RYaHUUASmU2DdV/O1sflfrZ3q8MLLWJSkmkZkuihMOdIxGgeBukxSovnQACaSmVsR6WOJiTZxFU0I7vyXF6FxWnEN356Vq5ezOApwAIdwDC6cQxWuoQZ1IPAEz/AKb1ZuvVjv1se0dcmazezDH1mfPynTlIA= JHEP10(2019)202 Figure 2. 5D tachyon solutions (in units of 1/L) for m̂2b = −1. LEFT: limit I with zIR = 1.2 zIR c . c 2 RIGHT: limit II with zIR = 20 zIR . We have taken λ = 1, and κ̂ = 1 (4) for the solid (dashed) line, and λ = −1, κ̂2 = 4 for the dotted line. ΛUV ∼ 1/zUV . The origin of the logarithm in eq. (3.20), that will play an important role, can be more easily understood by looking at the strongly-coupled dual theory; this has an explicit breaking of the conformal symmetry due to the double-trace marginal operator Og = |O∗ |2 that leads to a log-running of the couplings [38]. c , we can distinguish two Depending on the position of the IR-brane with respect to zIR limiting cases that will help to understand the physics of the model. These are illustrated in figure 2 and corresponds to c . In this case φL  1 for all z, meaning that the scale of confinement I) zIR ≈ zIR ∼ 1/zIR is larger than the scale of chiral breaking that is of order ∼ Lφ(zIR )/zIR . II) zIR  zIRc . In this case φL reaches O(1) values at some z = z  z , and then the χ IR scale of chiral breaking ∼ 1/zχ is larger than the scale of confinement ∼ 1/zIR . These two cases should be considered as formal limits, since in most of the parameter space of the model we will find that the IR-brane sits at zIR ∼ few × zIR c , i.e., between limits I and II, implying that naturally the scale of chiral breaking is similar to the scale of confinement. Let us start considering the limit I, where zIR is assumed to be just slightly above c . In this case the 4D mode φ (x) gets a small negative mass-squared, becoming a 4D zIR t tachyon. We find this mass is given by β zIR 4(m̂2b + 2)2 m2t ' − 2 ln c , where β= . (3.22) zIR zIR m̂4b + 6m̂2b + 10 Eq. (3.22) is only valid for |m2t |  1/zIR 2 that, obviously, requires a tuning in the parameter c  1 (that we will see later cannot be achieved by the radion space: either ln zIR /zIR minimization) or β  1 that requires m̂2b → −2. To find a stable configuration this 4D tachyon must have a positive quartic self-interaction, λt > 0, and this can arise either from λ or the feedback from gravity. We find λt = λ cλ (m̂2b ) + κ̂2 cκ (m̂2b ) , (3.23) – 10 – where cλ,κ are smooth and positive functions of m̂2b as derived in appendix B.1. The 4D tachyon VEV is then given by s 1 β zIR hφt i = ln c . (3.24) zIR λt zIR Therefore, in the limit I the z-profile of φ is given by eq. (3.20) with eq. (3.24) and N = LzIR /β, as we follow the normalization of φt of appendix B.1. This solution is shown in the left plot of figure 2 for λ = 1 and κ̂2 = 1, 4. In the limit κ̂2  1, we see from eqs. (3.20)–(3.24) that κ̂2 φ2 L2 stays constant and small. This means that the metric JHEP10(2019)202 remains always close to AdS5 . Let us now move to the limiting case II. First, let us neglect the feedback from the metric (κ̂2  1). As the IR-brane is now placed far away from zIR c , the tachyon profile grows ∝ z 2 till the quartic term of the potential becomes relevant. Solutions only exist if λ > 0, such that φ(z) settles q at the minimum of the 5D potential V (φ) where it takes the constant value φ(z) ' MΦ2 /λ = 2/(Lλ) (see right plot of figure 2).8 This means, in the c towards another CFT in which the dual interpretation, that the CFT flows at around 1/zIR global symmetry has been reduced to U(NF )V with Φs and Φa respectively transforming in the singlet and adjoint representation. In this new CFT, scale invariance is broken at a much lower scale 1/zIR . Let us now consider the feedback of the metric. For large κ̂2 the gravitational feedback becomes important before φ reaches the V (φ) minimum, making φ to enter into a “slow-roll” condition (see appendix A for details) delaying the position z at which φ gets its maximum ∼ 2/(Lλ). In this case λ < 0 is also possible as the slow-roll condition keeps φ(z) slowly growing till reaching the IR-brane (see right plot of figure 2). The metric evolves from AdS5 at z ≈ zIR c to another approximately AdS space at z  z c . 5 IR 3.2.2 Region m̂2b < −2 In this region we have that 2 + m̂2b is negative, and from the left-hand side of eq. (3.21), c is determined by the smallest zIR c zIR 1 ln '− , (3.25) zUV 2 + m̂2b that does not depend on . This means that non-trivial solutions for φ exist even if  < 0. These solutions however are supported by the IR-brane and for zIR → ∞ we have φ → 0. Therefore as soon as the IR-brane is not stabilized for  < 0 (i.e., zIR → ∞), we can also consider this region of the parameter space for studying the conformal transition. In this case the solution for φ, as we vary zIR , behaves in the following way. For c < z c0 c0 c0 √ zIR IR < zIR , where zIR is determined by ln(zIR /zUV ) ∼ π/ , we find that φ takes a nonzero value with a profile localized towards the IR-brane, φ ∼ (z/zIR )2 , as we said. The origin of this nonzero profile is that the IR-brane mass m̂2b , and not the 5D mass, is exceedingly negative. φ(zIR ) is mostly constant in this region and it does not help to 8 To satisfy the boundary condition at the IR-brane, φ must depart from 2/(Lλ) when approaching the IR-boundary, as can be appreciated in figure 2. – 11 – stabilize the IR-brane. On the other hand, for zIR > zIRc 0 , the profile of φ grows to become similar to the limit II discussed before (see right-hand side of figure 2), indicating that φ behaves as a genuine 5D tachyon. This latter behavior only occurs if the 5D mass is below the BF bound and can lead to a stable IR-brane. 3.3 Radion/dilaton stabilization Since the position of the IR-brane zIR is associated to a dynamical field, the radion (not necessary a mass eigenstate), its value must be determined dynamically. The extremization condition for zIR is exactly the junction condition eq. (3.18) after putting on-shell all other JHEP10(2019)202 fields. This can be written, using eq. (3.16) and eq. (3.17), as  s  2 2  02  Vb  6M5 1 + κ̂ L − V (φ) + Vb  = 0. (3.26) κ̂2 L2 L2 12 2M52 zIR For our particular case, this reduces to a quadratic equation for φ(zIR ): δ Λ̂2    4 2 2 1 2 2 λ̄ 4 δ Λ̂ − + κ̂ L m̄ φ (zIR ) − φ (zIR ) = 0 , (3.27) L2 12 2 4 where have introduced m̄2 ≡ [(m̂2b + 2)2 − 2δ Λ̂ m̂2b /3]/L2 , λ̄ ≡ λ + κ̂2 m̂4b /3, and L δ Λ̂ ≡ Λ4 + 6 , (3.28) M5 that is a measure of the detuning of the IR-brane tension away from the AdS5 value. Their values are bounded to be in the region 0 ≤ δ Λ̂ ≤ 6 . (3.29) The lower bound arises from demanding that for  < 0, the IR-brane is driven to zIR → ∞, such that the theory is in the AdS5 (CFT4 ) phase. From appendix B, in particular eq. (B.4), we see that δ Λ̂ is related to the self-coupling of the dilaton and δ Λ̂ ≥ 0 comes from re- quiring a positive dilaton self-coupling. On the other hand, the upper limit in eq. (3.29) is a more basic (geometrical) requirement: to possibly solve the junction condition even for dynamical solutions that start away from the minimum. If δ Λ̂ > 6, the IR-brane tension Λ4 is positive and it is easy to see that there would be no solutions where the IR-brane acts as an IR boundary (i.e., a cutoff of the AdS5 space at z = zIR ). Therefore these regions must be discarded. By playing with the parameters of the model, λ = ±1, κ̂2 , m̂2b and δ Λ̂, we can find regions where zIR is stabilized thanks to the presence of the 5D tachyon. These are shown in figure 3 in the plane m̂2b − δ Λ̂ for κ̂2 = 4 and λ = 1 (left plot), and λ = −1 (right plot). These regions are bounded from the left and the right at which, as it will be discussed later, the radion is massless. At the left boundary one obtains the lowest value of (a stabilized) zIR . For m̂2b > −2, this lowest value of the IR-brane position is achieved when δ Λ̂ = 0; for m̂2b → −2 we obtain the smallest zIR /zIR c that is given by z IR = e 1/2 z c , as can be IR – 12 – 6 6 0.1 0.1 5 5 0.2 0.2 0.6 0.3 4 4 0.3 AAAB+3icbVC7SgNBFJ2NrxhfayxtBoNgFXZF0DJoY2ERwTwgu4S7s7PJkNkHM3fFEPIrNhaK2Pojdv6Nk2QLTTwwcDjnHu6dE2RSaHScb6u0tr6xuVXeruzs7u0f2IfVtk5zxXiLpTJV3QA0lyLhLRQoeTdTHOJA8k4wupn5nUeutEiTBxxn3I9hkIhIMEAj9e2qF3KJ4A0BqXdnciH07ZpTd+agq8QtSI0UaPbtLy9MWR7zBJkErXuuk6E/AYWCST6teLnmGbARDHjP0ARirv3J/PYpPTVKSKNUmZcgnau/ExOItR7HgZmMAYd62ZuJ/3m9HKMrfyKSLEeesMWiKJcUUzorgoZCcYZybAgwJcytlA1BAUNTV8WU4C5/eZW0z+uu4fcXtcZ1UUeZHJMTckZcckka5JY0SYsw8kSeySt5s6bWi/VufSxGS1aROSJ/YH3+AIx3lBs= ˆ ⇤ 3 0.4 3 0.4 0.6 0.5 2 2 0.5 1 1 JHEP10(2019)202 0 0 -5 -4 -3 -2 -1 -5 -4 -3 -2 -1 m̂2b AAAB8XicbZBNS8NAEIYn9avWr6pHL4tF8FSSIuix6MVjBfuBbSyb7aZdutmE3YlQQv+FFw+KePXfePPfuG1z0NYXFh7emWFn3iCRwqDrfjuFtfWNza3idmlnd2//oHx41DJxqhlvsljGuhNQw6VQvIkCJe8kmtMokLwdjG9m9fYT10bE6h4nCfcjOlQiFIyitR56I4ok6gePtX654lbducgqeDlUIFejX/7qDWKWRlwhk9SYrucm6GdUo2CST0u91PCEsjEd8q5FRSNu/Gy+8ZScWWdAwljbp5DM3d8TGY2MmUSB7YwojsxybWb+V+umGF75mVBJilyxxUdhKgnGZHY+GQjNGcqJBcq0sLsSNqKaMrQhlWwI3vLJq9CqVT3LdxeV+nUeRxFO4BTOwYNLqMMtNKAJDBQ8wyu8OcZ5cd6dj0VrwclnjuGPnM8fzcuQVQ== m̂2b AAAB8XicbZBNS8NAEIYn9avWr6pHL4tF8FSSIuix6MVjBfuBbSyb7aZdutmE3YlQQv+FFw+KePXfePPfuG1z0NYXFh7emWFn3iCRwqDrfjuFtfWNza3idmlnd2//oHx41DJxqhlvsljGuhNQw6VQvIkCJe8kmtMokLwdjG9m9fYT10bE6h4nCfcjOlQiFIyitR56I4ok6gePtX654lbducgqeDlUIFejX/7qDWKWRlwhk9SYrucm6GdUo2CST0u91PCEsjEd8q5FRSNu/Gy+8ZScWWdAwljbp5DM3d8TGY2MmUSB7YwojsxybWb+V+umGF75mVBJilyxxUdhKgnGZHY+GQjNGcqJBcq0sLsSNqKaMrQhlWwI3vLJq9CqVT3LdxeV+nUeRxFO4BTOwYNLqMMtNKAJDBQ8wyu8OcZ5cd6dj0VrwclnjuGPnM8fzcuQVQ== Figure 3. Region of the parameter space that leads to a stable IR-brane for κ̂2 = 4 and λ = 1 (left) and λ = −1 (right). We also provide the value of the lightest scalar mass, mS1 /mρ . analytically found by looking at the 4D effective theory (see appendix B). On the other hand, as we get close to the boundary on the right of the regions in figure 3, we have zIR → ∞ (limit II). In most of the colored regions however we have that zIR ∼ zχ . In other words, the model naturally predicts the scale of chiral symmetry breaking to be around the scale of confinement. If the radion is the lightest 4D mode in the theory, we can use eq. (3.26) to obtain its effective potential. The radion corresponds in the dual 4D CFT to the dilaton, φd , whose VEV determines the scales of the model. For this reason φd at the minimum is related with the warp factor evaluated at z = zIR . Nevertheless, outside the minimum eq. (3.26) the relation of zIR with φd is a more complicated function, zIR = f (φd ), especially in the basis where φd is canonically normalized. Going off-shell requires not equating the l.h.s. of eq. (3.26) to zero, and identifying this with the first derivative of the dilaton effective potential: s  3 2 4  02  2 3 dVeff (φd ) φd  κ̂ L V b κ̂ L = 1+ − V (φ) + Vb  , (3.30) dφd n(φd ) 12 2M52 6M5 zIR =f (φd ) where n(φd ) > 0 is in general a complicated function of φd (that cannot be zero, otherwise we will have an extra minimum beyond eq. (3.26)) that we do not need to specify here. By integrating eq. (3.30) over φd , one can obtain the dilaton effective potential Veff (φd ). For the simple case in which the backreaction is neglected and the space is just AdS 5 , we have zIR ∝ 1/φd and n(φd ) is just a constant. For this case we show the effective potential (up to an overall constant) in figure 4. We can see that the potential has a minimum at c and goes to a constant value at large z , where φ becomes constant as zIR ' 2 − 3 zIR IR it approaches the minimum of its 5D potential. For a better understanding of the dilaton effective potential, we can analytically calculate the effective potential of the 4D tachyon – 13 – 0 m̂2b = AAAB+nicbZDLSsNAFIZPvNZ6S3XpZrAIbixJKehGKLpxWcFeoI1hMp20QyeTMDNRSuyjuHGhiFufxJ1v47TNQlt/GPj4zzmcM3+QcKa043xbK6tr6xubha3i9s7u3r5dOmipOJWENknMY9kJsKKcCdrUTHPaSSTFUcBpOxhdT+vtByoVi8WdHifUi/BAsJARrI3l26XeEGsU3Vf9LJhcnrmVmm+XnYozE1oGN4cy5Gr49levH5M0okITjpXquk6ivQxLzQink2IvVTTBZIQHtGtQ4IgqL5udPkEnxumjMJbmCY1m7u+JDEdKjaPAdEZYD9VibWr+V+umOrzwMiaSVFNB5ovClCMdo2kOqM8kJZqPDWAimbkVkSGWmGiTVtGE4C5+eRla1Ypr+LZWrl/lcRTgCI7hFFw4hzrcQAOaQOARnuEV3qwn68V6tz7mrStWPnMIf2R9/gBfl5LB 1.4 m̂2b = 1.2 -0.01 Ve↵ AAAB+nicbZDLSsNAFIZP6q3WW6pLN4NFcGNJiqAboejGZQV7gTaGyXTSDp1MwsxEKbGP4saFIm59Ene+jdM2C239YeDjP+dwzvxBwpnSjvNtFVZW19Y3ipulre2d3T27vN9ScSoJbZKYx7ITYEU5E7Spmea0k0iKo4DTdjC6ntbbD1QqFos7PU6oF+GBYCEjWBvLt8u9IdYouq/5WTC5PHWrNd+uOFVnJrQMbg4VyNXw7a9ePyZpRIUmHCvVdZ1EexmWmhFOJ6VeqmiCyQgPaNegwBFVXjY7fYKOjdNHYSzNExrN3N8TGY6UGkeB6YywHqrF2tT8r9ZNdXjhZUwkqaaCzBeFKUc6RtMcUJ9JSjQfG8BEMnMrIkMsMdEmrZIJwV388jK0alXX8O1ZpX6Vx1GEQziCE3DhHOpwAw1oAoFHeIZXeLOerBfr3fqYtxasfOYA/sj6/AFcj5K/ AAAB8nicbZBNSwMxEIZn61etX1WPXoJF8FR2RdBj0YvHCrYW2qVk09k2NNksSVYoS3+GFw+KePXXePPfmLZ70NYXAg/vzJCZN0oFN9b3v73S2vrG5lZ5u7Kzu7d/UD08ahuVaYYtpoTSnYgaFDzBluVWYCfVSGUk8DEa387qj0+oDVfJg52kGEo6THjMGbXO6rb7eU9LgnE87Vdrft2fi6xCUEANCjX71a/eQLFMYmKZoMZ0Az+1YU615UzgtNLLDKaUjekQuw4TKtGE+XzlKTlzzoDESruXWDJ3f0/kVBozkZHrlNSOzHJtZv5X62Y2vg5znqSZxYQtPoozQawis/vJgGtkVkwcUKa525WwEdWUWZdSxYUQLJ+8Cu2LeuD4/rLWuCniKMMJnMI5BHAFDbiDJrSAgYJneIU3z3ov3rv3sWgtecXMMfyR9/kDHTaRIQ== -0.02 m̂2b = AAAB+HicbZDLSgMxFIbPeK310lGXboJFcGOZKYJuhKIblxXsBdpxyKRpG5pkhiQj1KFP4saFIm59FHe+jWk7C239IfDxn3M4J3+UcKaN5307K6tr6xubha3i9s7uXsndP2jqOFWENkjMY9WOsKacSdowzHDaThTFIuK0FY1upvXWI1WaxfLejBMaCDyQrM8INtYK3VJ3iA0SD9UwiyZXZ37olr2KNxNaBj+HMuSqh+5XtxeTVFBpCMdad3wvMUGGlWGE00mxm2qaYDLCA9qxKLGgOshmh0/QiXV6qB8r+6RBM/f3RIaF1mMR2U6BzVAv1qbmf7VOavqXQcZkkhoqyXxRP+XIxGiaAuoxRYnhYwuYKGZvRWSIFSbGZlW0IfiLX16GZrXiW747L9eu8zgKcATHcAo+XEANbqEODSCQwjO8wpvz5Lw4787HvHXFyWcO4Y+czx92fpJL sha1_base64="ck8pdC+ekZH4nUmSP+ZG7r8lEyk=">AAAB2XicbZDNSgMxFIXv1L86Vq1rN8EiuCozbnQpuHFZwbZCO5RM5k4bmskMyR2hDH0BF25EfC93vo3pz0JbDwQ+zknIvSculLQUBN9ebWd3b/+gfugfNfzjk9Nmo2fz0gjsilzl5jnmFpXU2CVJCp8LgzyLFfbj6f0i77+gsTLXTzQrMMr4WMtUCk7O6oyaraAdLMW2IVxDC9YaNb+GSS7KDDUJxa0dhEFBUcUNSaFw7g9LiwUXUz7GgUPNM7RRtRxzzi6dk7A0N+5oYkv394uKZ9bOstjdzDhN7Ga2MP/LBiWlt1EldVESarH6KC0Vo5wtdmaJNChIzRxwYaSblYkJN1yQa8Z3HYSbG29D77odOn4MoA7ncAFXEMIN3MEDdKALAhJ4hXdv4r15H6uuat66tDP4I+/zBzjGijg= sha1_base64="QwvF8UUtSyH7vpr2KltYOxbKEY0=">AAAB7XicbZBLSwMxFIXv+Ky12tGtm2AR3FhmutGNILhxWcE+oB2HTJppQ5OZIbkj1KG/xI0LRfw77vw3po+Fth4IfJyTcG9OlElh0PO+nY3Nre2d3dJeeb9ycFh1jyptk+aa8RZLZaq7ETVcioS3UKDk3UxzqiLJO9H4dpZ3nrg2Ik0ecJLxQNFhImLBKFordKv9EUWiHhthEU2vL/zQrXl1by6yDv4SarBUM3S/+oOU5YonyCQ1pud7GQYF1SiY5NNyPzc8o2xMh7xnMaGKm6CYLz4lZ9YZkDjV9iRI5u7vFwVVxkxUZG8qiiOzms3M/7JejvFVUIgky5EnbDEoziXBlMxaIAOhOUM5sUCZFnZXwkZUU4a2q7ItwV/98jq0G3Xf8r0HJTiBUzgHHy7hBu6gCS1gkMMLvMG78+y8Oh+LujacZW/H8EfO5w8qU5Do sha1_base64="BQ6Ewj/QCpJrGML0rJyjqDw6I/M=">AAAB+HicbZDLSgMxFIbP1Futl466dBMsghvLTDe6EYpuXFawF2jHIZOmbWiSGZKMUIc+iRsXirj1Udz5NqbtLLT1h8DHf87hnPxRwpk2nvftFNbWNza3itulnd29/bJ7cNjScaoIbZKYx6oTYU05k7RpmOG0kyiKRcRpOxrfzOrtR6o0i+W9mSQ0EHgo2YARbKwVuuXeCBskHmphFk2vzv3QrXhVby60Cn4OFcjVCN2vXj8mqaDSEI617vpeYoIMK8MIp9NSL9U0wWSMh7RrUWJBdZDND5+iU+v00SBW9kmD5u7viQwLrScisp0Cm5Fers3M/2rd1Awug4zJJDVUksWiQcqRidEsBdRnihLDJxYwUczeisgIK0yMzapkQ/CXv7wKrVrVt3znVerXeRxFOIYTOAMfLqAOt9CAJhBI4Rle4c15cl6cd+dj0Vpw8pkj+CPn8wd1PpJH 1 2 3 4 5 10 c zIR /zIR JHEP10(2019)202 AAACBHicbZC7TsMwFIadcivlFmDsYlEhMZUEIcFYwQJbQfQitSFyXKe1ajuR7SCVKAMLr8LCAEKsPAQbb4PbZigtv2Tp83/OkX3+IGZUacf5sQpLyyura8X10sbm1vaOvbvXVFEiMWngiEWyHSBFGBWkoalmpB1LgnjASCsYXo7rrQciFY3EnR7FxOOoL2hIMdLG8u1y+uinXcnh9W2WHc9c7nHm2xWn6kwEF8HNoQJy1X37u9uLcMKJ0JghpTquE2svRVJTzEhW6iaKxAgPUZ90DArEifLSyRIZPDROD4aRNEdoOHFnJ1LElRrxwHRypAdqvjY2/6t1Eh2eeykVcaKJwNOHwoRBHcFxIrBHJcGajQwgLKn5K8QDJBHWJreSCcGdX3kRmidV1/DNaaV2kcdRBGVwAI6AC85ADVyBOmgADJ7AC3gD79az9Wp9WJ/T1oKVz+yDP7K+fgHEkpgn Figure 4. Dilaton effective potential for λ = 1, κ̂2 = 0, δ Λ̂ = 0 and different values of m̂2b . c ≈ 1. This is done in appendix B. This shows that the and dilaton in the limit zIR /zIR origin of the existence of a minimum in Veff (φd ) can be tracked back to the log-dependence in eq. (3.22). 3.4 Excitations around the 5D tachyon The main interest of the article is to know whether close to the conformal transition there is a light dilaton, as often claimed in the literature. Therefore we will start considering the flavor-singlet 0++ spectrum of the theory, to analyze later other sectors. 3.4.1 The singlet scalar sector and light dilaton The flavor-singlet 0++ spectrum is composed by the radion (the only scalar in the gravi- tational sector) and the excitations of Φs around the background φ(z). Since the mixing of the dilaton with Φs is in principle sizable for NF ∼ Nc (κ̂2 ∼ 1), we must consider the coupled EOM between the scalar sector and the gravitational sector. The equations for the mass spectrum are given in appendix A.2 and must be solved numerically. For the lightest mode S1 the results are presented in figure 3 for λ = ±1. We have normalized the S1 mass to the one of the lightest vector resonance, mρ , being this latter the lightest state in real QCD and holographic versions [33–35]. Figure 3 shows that the 0++ state is always lighter than the vector in all regions of the parameter space. At the boundary of the regions at which IR-brane stabilization is achieved, the dilaton is massless, but its mass is roughly below half of the ρ mass in most of the interior region. We have checked that for larger κ̂2 , the value of mS1 increases but not significantly. To understand why the dilaton mass is small, it is convenient to show how its mass varies as a function of zIR for different values of m̂2b . This corresponds to moving in vertical lines in the plane of figure 3 from the bottom to the top, trading the parameter δ Λ̂ for zIR by means of eq. (3.27). We remark however that we will present results for a wide region of zIR , going beyond the allowed region eq. (3.29). This will help us to understand the origin of the smallness of mS1 /mρ . The result is shown in figure 5 for different values of – 14 – 0.6 0.6 0.5 AAAB8HicbVDLSgMxFL1TX7W+qi7dBIvgqsyIoBuh6MZlBVsr7VAymUwbmseQZIQy9CvcuFDErZ/jzr8xbWehrQcCh3POJfeeKOXMWN//9korq2vrG+XNytb2zu5edf+gbVSmCW0RxZXuRNhQziRtWWY57aSaYhFx+hCNbqb+wxPVhil5b8cpDQUeSJYwgq2THnvcRWN8FfSrNb/uz4CWSVCQGhRo9qtfvViRTFBpCcfGdAM/tWGOtWWE00mllxmaYjLCA9p1VGJBTZjPFp6gE6fEKFHaPWnRTP09kWNhzFhELimwHZpFbyr+53Uzm1yGOZNpZqkk84+SjCOr0PR6FDNNieVjRzDRzO2KyBBrTKzrqOJKCBZPXibts3rg+N15rXFd1FGGIziGUwjgAhpwC01oAQEBz/AKb572Xrx372MeLXnFzCH8gff5AzMnj/8= =1 0.5 AAAB8XicbVDLSgMxFL1TX7W+qi7dBIvgxjIjgm6EohuXFewD26FkMrdtaCYzJBmhDP0LNy4UcevfuPNvTNtZaOuBwOGcc8m9J0gE18Z1v53Cyura+kZxs7S1vbO7V94/aOo4VQwbLBaxagdUo+ASG4Ybge1EIY0Cga1gdDv1W0+oNI/lgxkn6Ed0IHmfM2qs9NgVNhrS6zOvV664VXcGsky8nFQgR71X/uqGMUsjlIYJqnXHcxPjZ1QZzgROSt1UY0LZiA6wY6mkEWo/m208ISdWCUk/VvZJQ2bq74mMRlqPo8AmI2qGetGbiv95ndT0r/yMyyQ1KNn8o34qiInJ9HwScoXMiLEllCludyVsSBVlxpZUsiV4iycvk+Z51bP8/qJSu8nrKMIRHMMpeHAJNbiDOjSAgYRneIU3RzsvzrvzMY8WnHzmEP7A+fwBn3aQNg== = 1 0.4 m̂2b = 1.5 0.4 mS1 AAAB+HicbZDLSgMxFIbPeK310lGXboJFcGOZKYpuhKIblxXsBdpxyKSZNjTJDElGqKVP4saFIm59FHe+jekF0dYfAh//OYdz8kcpZ9p43peztLyyurae28hvbm3vFNzdvbpOMkVojSQ8Uc0Ia8qZpDXDDKfNVFEsIk4bUf96XG88UKVZIu/MIKWBwF3JYkawsVboFto9bJC4L4fR5YlfOgvdolfyJkI/4M9DEWaqhu5nu5OQTFBpCMdat3wvNcEQK8MIp6N8O9M0xaSPu7RlUWJBdTCcHD5CR9bpoDhR9kmDJu7viSEWWg9EZDsFNj09Xxub/9VamYkvgiGTaWaoJNNFccaRSdA4BdRhihLDBxYwUczeikgPK0yMzSpvQ1j48iLUyyXf8u1psXI1iyMHB3AIx+DDOVTgBqpQAwIZPMELvDqPzrPz5rxPW5ec2cw+/JHz8Q2T8ZG3 0.3 0.3 m⇢ m̂2b = AAAB9HicbZBNSwMxEIZn/az1q+rRS7AIXiy7RdCLUPTisYL9gHYt2TTbhmazazJbKKW/w4sHRbz6Y7z5b0zbPWjrC4GHd2aYyRskUhh03W9nZXVtfWMzt5Xf3tnd2y8cHNZNnGrGayyWsW4G1HApFK+hQMmbieY0CiRvBIPbab0x5NqIWD3gKOF+RHtKhIJRtJbf7lMk0WO5E1yfe51C0S25M5Fl8DIoQqZqp/DV7sYsjbhCJqkxLc9N0B9TjYJJPsm3U8MTyga0x1sWFY248cezoyfk1DpdEsbaPoVk5v6eGNPImFEU2M6IYt8s1qbmf7VWiuGVPxYqSZErNl8UppJgTKYJkK7QnKEcWaBMC3srYX2qKUObU96G4C1+eRnq5ZJn+f6iWLnJ4sjBMZzAGXhwCRW4gyrUgMETPMMrvDlD58V5dz7mrStONnMEf+R8/gAyZ5EO 1 AAAB/3icbZDLSsNAFIYn9VbrLSq4cTNYBFclEUGXRTcuK9paaEKYTCft0LmEmYlQYha+ihsXirj1Ndz5Nk7bLLT1h4GP/5zDOfPHKaPaeN63U1laXlldq67XNja3tnfc3b2OlpnCpI0lk6obI00YFaRtqGGkmyqCeMzIfTy6mtTvH4jSVIo7M05JyNFA0IRiZKwVuQdBohDOeZTfRn5RWAjUUBaRW/ca3lRwEfwS6qBUK3K/gr7EGSfCYIa07vleasIcKUMxI0UtyDRJER6hAelZFIgTHebT+wt4bJ0+TKSyTxg4dX9P5IhrPeax7eTIDPV8bWL+V+tlJrkIcyrSzBCBZ4uSjEEj4SQM2KeKYMPGFhBW1N4K8RDZQIyNrGZD8Oe/vAid04Zv+eas3rws46iCQ3AEToAPzkETXIMWaAMMHsEzeAVvzpPz4rw7H7PWilPO7IM/cj5/ALMeloc= 0.2 0.2 m̂2b = 0.5 0.1 0.1 AAAB+HicbZDLSsNAFIZP6q3WS6Mu3QwWwY0lKYpuhKIblxXsBdoYJtNJO3QyCTMToYY+iRsXirj1Udz5Nk7bLLT1h4GP/5zDOfMHCWdKO863VVhZXVvfKG6WtrZ3dsv23n5LxakktEliHstOgBXlTNCmZprTTiIpjgJO28HoZlpvP1KpWCzu9TihXoQHgoWMYG0s3y73hlij6KHmB1enTvXctytO1ZkJLYObQwVyNXz7q9ePSRpRoQnHSnVdJ9FehqVmhNNJqZcqmmAywgPaNShwRJWXzQ6foGPj9FEYS/OERjP390SGI6XGUWA6I6yHarE2Nf+rdVMdXnoZE0mqqSDzRWHKkY7RNAXUZ5ISzccGMJHM3IrIEEtMtMmqZEJwF7+8DK1a1TV8d1apX+dxFOEQjuAEXLiAOtxCA5pAIIVneIU368l6sd6tj3lrwcpnDuCPrM8fkP2RtQ== 0.0 0.0 0 10 20 30 40 5 10 15 20 25 30 35 40 c c zIR /zIR zIR /zIR JHEP10(2019)202 AAACBHicbZC7TsMwFIadcivlFmDsYlEhMZUEIcFYwQJbQfQitSFyXKe1ajuR7SCVKAMLr8LCAEKsPAQbb4PbZigtv2Tp83/OkX3+IGZUacf5sQpLyyura8X10sbm1vaOvbvXVFEiMWngiEWyHSBFGBWkoalmpB1LgnjASCsYXo7rrQciFY3EnR7FxOOoL2hIMdLG8u1y+uinXcnh9W2WHc9c7nHm2xWn6kwEF8HNoQJy1X37u9uLcMKJ0JghpTquE2svRVJTzEhW6iaKxAgPUZ90DArEifLSyRIZPDROD4aRNEdoOHFnJ1LElRrxwHRypAdqvjY2/6t1Eh2eeykVcaKJwNOHwoRBHcFxIrBHJcGajQwgLKn5K8QDJBHWJreSCcGdX3kRmidV1/DNaaV2kcdRBGVwAI6AC85ADVyBOmgADJ7AC3gD79az9Wp9WJ/T1oKVz+yDP7K+fgHEkpgn AAACBHicbZC7TsMwFIadcivlFmDsYlEhMZUEIcFYwQJbQfQitSFyXKe1ajuR7SCVKAMLr8LCAEKsPAQbb4PbZigtv2Tp83/OkX3+IGZUacf5sQpLyyura8X10sbm1vaOvbvXVFEiMWngiEWyHSBFGBWkoalmpB1LgnjASCsYXo7rrQciFY3EnR7FxOOoL2hIMdLG8u1y+uinXcnh9W2WHc9c7nHm2xWn6kwEF8HNoQJy1X37u9uLcMKJ0JghpTquE2svRVJTzEhW6iaKxAgPUZ90DArEifLSyRIZPDROD4aRNEdoOHFnJ1LElRrxwHRypAdqvjY2/6t1Eh2eeykVcaKJwNOHwoRBHcFxIrBHJcGajQwgLKn5K8QDJBHWJreSCcGdX3kRmidV1/DNaaV2kcdRBGVwAI6AC85ADVyBOmgADJ7AC3gD79az9Wp9WJ/T1oKVz+yDP7K+fgHEkpgn Figure 5. Masses of the lightest singlet scalar S1 normalized to the vector mass. We have taken κ̂2 = 1 (solid line), κ̂2 = 4 (dashed line) and κ̂2 = 10 (dot-dashed line). LEFT: λ = 1 and m̂2b = −1.5 (−0.5) for the upper blue (lower black) lines. RIGHT: λ = −1 and m̂2b = −1. m̂2b > −2. We see that the dilaton mass starts at zero at zIR ' zIR c , grows for intermediate c , and tends again to zero for z /z c → ∞. zIR /zIR IR IR We can understand this behaviour analytically. Assuming that S1 is the radion/dilaton, we can analytically obtain its mass by taking the derivative of eq. (3.30) evaluated at the minimum eq. (3.26). We obtain a2 Vb0 Vb00 Vb0     m2φd ' 4κ̂2 a(zIR )L2 −V0 − ∂zIR φ(zIR ) , (3.31) 24ȧ M52 6M5 zIR where we have used that at the minimum φ3d ∂φd f (φd )/n(φd ) ' −4a(zIR )/L derived in appendix A. In our particular case eq. (3.31) reduces, after normalizing to the vector mass eq. (3.39), to m2φd ' P (φ(zIR )) Q(φ(zIR )) βφ (zIR ) , (3.32) m2ρ where we have defined the dimensionless functions κ̂2 φ2 (zIR ) 8κ̂2 L2 φ2 (zIR ) P (φ(zIR )) ≡ ' , (3.33) 6m2ρ 27π 2 Ȧ2IR 2 m̂b =0 1 h 2 i Q(φ(zIR )) ≡ λφ (zIR )L2 − (m̂2b + 2)2 − 4m̂2b (ȦIR − 1) , (3.34) ȦIR L ∂zIR φ(zIR ) βφ (zIR ) ≡ , (3.35) a(zIR ) φ(zIR ) with s κ̂2 L2 φ2 λL2 2   ȧL ȦIR ≡− 2 = 1+ 4+ m̂4b − φ , ȦIR ≥ 1 , (3.36) a zIR 24 2 zIR – 15 – where we have used eq. (3.16). From eq. (3.32) we can infer different regimes at which the dilaton can be light: • The prefactor P (φ(zIR )) is suppressed for κ̂2 φ2 (zIR )  1/L2 . Therefore in the limit I the dilaton is always light, even when formally we take κ̂2  1 (see discussion after eq. (3.24)). Also for large values of Lφ, possible in the limit II with λ < 0, we have P → 1/(Lφ)2 , and consequently the dilaton mass is suppressed. • The function Q(φ(zIR )) determines the sign of m2φd . In the limit I we have φL → 0 and ȦIR → 1, and then Q becomes negative. This means that the dilaton effective JHEP10(2019)202 potential has actually no minimum, as we already pointed out in section 3.3. As we c , Q increases till becoming zero, corresponding to the points seen in increase zIR /zIR figure 5 with mφd = 0. One can check that they are inflection points of the dilaton potential. • The function βφ is the main responsible for natural light dilatons in Goldberger-Wise models [10–14]. Since moving simultaneously the UV and IR boundaries does not change physical quantities, we can deduce a(zIR ) ∂zIR φ(zIR ) = − ∂z φ(zIR ) , (3.37) a(zUV ) UV showing that βφ is in fact sensitive to the dependence of the tachyon φ with variations of the UV boundary, and therefore to the explicit breaking of conformal invariance (that arises due to the presence of the UV cutoff). For this reason βφ is directly related with the beta function βλeff of the dilaton effective coupling of eq. (1.1). βφ explains why the dilaton mass always goes to zero for large zIR /zIR c . Indeed, as we approach the limit II for λ > 0, the 5D tachyon goes to the minimum of its potential where it becomes constant. We then expect βφ  1. Also in the limit II for λ < 0 the slow-roll conditions are achieved and βφ tends to zero. Unfortunately, these regions of a parametrically light dilaton are very small in the full parameter space of the model, c requires an adjustment see figure 3, since stabilizing the IR-brane at large zIR /zIR of the parameters of the model. In the limit I we can derive from eq. (3.20) and c )), and using the eq. (B.5) we get β ∼ β that is eq. (3.24) that βφ ∼ 1/(2 ln(zIR /zIR φ nonzero but smallish in the regions considered.9 The situation is similar in regions with m̂2b < −2 (see figure 3 or figure 8). The only main difference is that the dilaton mass goes to zero for small values of zIR /zIR c 0 , not due to Q → 0, but because φ(zIR ) tends to a constant value as explained in section 3.2.2, and therefore βφ → 0. We conclude that the dilaton mass is parametrically smaller than mρ at small and large zIR (respectively corresponding to the left and right boundaries of the regions of figure 3). At small zIR , the reason is either the existence of an inflection point in the dilaton potential 9 Notice that ∂zIR φ(zIR ) 6= ∂z φ|zIR , and then βφ does not measure the growth of the tachyon (that is power-law ∼ z 2 ), but its variation as we move the IR-brane (or UV-boundary) that it is much milder (logarithmic). – 16 – (for the case m̂2b > −2) or that φ(zIR ) becomes frozen and βφ → 0 (for the case m̂2b < −2). Also at small 5D tachyon values its log-dependence on zIR gives a smallish βφ and therefore a smallish dilaton mass. At large zIR (limit II) the geometry approaches again AdS5 (the dual model flows towards another approximate CFT4 ) where scale invariance is partially recovered and therefore the dilaton mass must go to zero. “Trapped” between these two limits, the dilaton mass cannot grow much in the intermediate region and then remains always the lightest resonance (although not parametrically lighter than the others). Finally, we would also like to discuss the mass of the second lightest singlet scalar, S2 . This is also obtained numerically (see appendix A.2), and the result is shown in figure 6 for JHEP10(2019)202 certain representative values of the parameter space. This scalar S2 is mostly the excitation around the profile φ(z) (up to a small mixing with the radion), a Higgs-like state. For this reason when zIR → zIR c , we expect m2 → λ φ2 → 0, as appreciated in figure 6. S2 t t 3.4.2 Non-singlet scalars, vector and axial-vector excitations For the scalars in the adjoint under the SU(NF )V symmetry, Φa , the EOM is given by [∂µ2 − a−3 ∂z a3 ∂z + a2 MΦ2 + a2 (3λ − 2NF λ2 )φ2 (z)]Φa = 0 . (3.38) As we already mentioned, there are two important difference with respect the singlet scalar case. First, the scalars in the adjoint do not mix with the radion/dilaton. Second, the quartic coupling in eq. (3.38) is different from the singlet case due to the presence of λ2 . This implies that the adjoint scalar masses are expected to be different from the singlet scalar masses, with the magnitude of the mass splitting being sensitive to κ̂2 and λ2 . √ We are also interested in the vector VM = (LM + RM )/ 2 and axial-vector √ AM = (LM − RM )/ 2 spectrum [33–35]. The vector spectrum is only indirectly sensitive to the tachyon through its impact to the metric. Therefore, since flavor-singlet resonances (the ω in QCD) and adjoint resonances (the ρ in QCD) feel the same metric and have the same boundary conditions, they get equal masses. This is an important prediction of the 5D model.10 It is useful to have an approximate analytic value for mρ , since we are using this mass to normalize the other resonance masses. This is possible in the limit in which the 5D space is approximately AdS, that corresponds to limits I and II, as we explained in section 3.2. In AdS5 we have mρ ∼ −(3π/4)(ȧ/a). We find that a reasonably good approximation for ȧ/a in the limit of small and large zIR is given by eq. (3.16) neglecting the derivative terms and taking φ at z = zIR .11 We then have: 3π a(zIR ) mρ ' ȦIR . (3.39) 4 L m̂2b =0 We have checked that this value is within . 20% the exact mass of ρ for the regions of the parameter space studied in this article. 10 Of course, mass splittings could be generated at the loop level or from higher-dimensional operators in eq. (3.2), but these are expected to be suppressed. 11 We put to zero the derivative terms to avoid the drastic change of φ near the IR boundary -see footnote 8. – 17 – 2.0 a0 AAAB6nicbZBNS8NAEIYn9avWr6hHL4tF8FQSEfRY9OKxov2ANpTNdtMu3WzC7kQooT/BiwdFvPqLvPlv3LY5aOsLCw/vzLAzb5hKYdDzvp3S2vrG5lZ5u7Kzu7d/4B4etUySacabLJGJ7oTUcCkUb6JAyTup5jQOJW+H49tZvf3EtRGJesRJyoOYDpWIBKNorQfa9/pu1at5c5FV8AuoQqFG3/3qDRKWxVwhk9SYru+lGORUo2CSTyu9zPCUsjEd8q5FRWNugny+6pScWWdAokTbp5DM3d8TOY2NmcSh7YwpjsxybWb+V+tmGF0HuVBphlyxxUdRJgkmZHY3GQjNGcqJBcq0sLsSNqKaMrTpVGwI/vLJq9C6qPmW7y+r9ZsijjKcwCmcgw9XUIc7aEATGAzhGV7hzZHOi/PufCxaS04xcwx/5Hz+AOcVjYg= 1.5 a1 AAAB6nicbZBNS8NAEIYn9avWr6hHL4tF8FQSEfRY9OKxov2ANpTJdtMu3WzC7kYooT/BiwdFvPqLvPlv3LY5aOsLCw/vzLAzb5gKro3nfTultfWNza3ydmVnd2//wD08aukkU5Q1aSIS1QlRM8ElaxpuBOukimEcCtYOx7ezevuJKc0T+WgmKQtiHEoecYrGWg/Y9/tu1at5c5FV8AuoQqFG3/3qDRKaxUwaKlDrru+lJshRGU4Fm1Z6mWYp0jEOWdeixJjpIJ+vOiVn1hmQKFH2SUPm7u+JHGOtJ3FoO2M0I71cm5n/1bqZia6DnMs0M0zSxUdRJohJyOxuMuCKUSMmFpAqbncldIQKqbHpVGwI/vLJq9C6qPmW7y+r9ZsijjKcwCmcgw9XUIc7aEATKAzhGV7hzRHOi/PufCxaS04xcwx/5Hz+AOiZjYk= mi 1.0 m⇢ F⇡ AAAB+3icbZDLSgMxFIbPeK31VuvSTbAIrsqMCLosunFZwV6gMwyZNNOGJpMhyYhlmFdx40IRt76IO9/GtJ2Fth5I+Pj/c8jJH6WcaeO6387a+sbm1nZlp7q7t39wWDuqd7XMFKEdIrlU/QhryllCO4YZTvupolhEnPaiye3M7z1SpZlMHsw0pYHAo4TFjGBjpbBW92OFSS5CVtjLV2NZhLWG23TnhVbBK6EBZbXD2pc/lCQTNDGEY60HnpuaIMfKMMJpUfUzTVNMJnhEBxYTLKgO8vnuBTqzyhDFUtmTGDRXf0/kWGg9FZHtFNiM9bI3E//zBpmJr4OcJWlmaEIWD8UZR0aiWRBoyBQlhk8tYKKY3RWRMbZhGBtX1YbgLX95FboXTc/y/WWjdVPGUYETOIVz8OAKWnAHbegAgSd4hld4cwrnxXl3Phata045cwx/yvn8Ac+3lO0= S2 AAAB6nicbZBNS8NAEIYn9avWr6pHL4tF8FSSIuix6MVjpfYD2lA22027dLMJuxOhhP4ELx4U8eov8ua/cdvmoK0vLDy8M8POvEEihUHX/XYKG5tb2zvF3dLe/sHhUfn4pG3iVDPeYrGMdTeghkuheAsFSt5NNKdRIHknmNzN650nro2I1SNOE+5HdKREKBhFazWbg9qgXHGr7kJkHbwcKpCrMSh/9YcxSyOukElqTM9zE/QzqlEwyWelfmp4QtmEjnjPoqIRN362WHVGLqwzJGGs7VNIFu7viYxGxkyjwHZGFMdmtTY3/6v1Ugxv/EyoJEWu2PKjMJUEYzK/mwyF5gzl1AJlWthdCRtTTRnadEo2BG/15HVo16qe5YerSv02j6MIZ3AOl+DBNdThHhrQAgYjeIZXeHOk8+K8Ox/L1oKTz5zCHzmfP9TJjXw= AAAB7HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfRYFMRjBdMW2lA22027dLMJuxOhhP4GLx4U8eoP8ua/cdvmoK0vLDy8M8POvGEqhUHX/XZKa+sbm1vl7crO7t7+QfXwqGWSTDPus0QmuhNSw6VQ3EeBkndSzWkcSt4Ox7ezevuJayMS9YiTlAcxHSoRCUbRWv5dv5eKfrXm1t25yCp4BdSgULNf/eoNEpbFXCGT1Jiu56YY5FSjYJJPK73M8JSyMR3yrkVFY26CfL7slJxZZ0CiRNunkMzd3xM5jY2ZxKHtjCmOzHJtZv5X62YYXQe5UGmGXLHFR1EmCSZkdjkZCM0ZyokFyrSwuxI2opoytPlUbAje8smr0Lqoe5YfLmuNmyKOMpzAKZyDB1fQgHtogg8MBDzDK7w5ynlx3p2PRWvJKWaO4Y+czx+V6Y6G 0.5 S1 JHEP10(2019)202 0.0 AAAB6nicbZBNS8NAEIYn9avWr6pHL4tF8FQSEeqx6MVjpfYD2lA22027dLMJuxOhhP4ELx4U8eov8ua/cdvmoK0vLDy8M8POvEEihUHX/XYKG5tb2zvF3dLe/sHhUfn4pG3iVDPeYrGMdTeghkuheAsFSt5NNKdRIHknmNzN650nro2I1SNOE+5HdKREKBhFazWbA29QrrhVdyGyDl4OFcjVGJS/+sOYpRFXyCQ1pue5CfoZ1SiY5LNSPzU8oWxCR7xnUdGIGz9brDojF9YZkjDW9ikkC/f3REYjY6ZRYDsjimOzWpub/9V6KYY3fiZUkiJXbPlRmEqCMZnfTYZCc4ZyaoEyLeyuhI2ppgxtOiUbgrd68jq0r6qe5YfrSv02j6MIZ3AOl+BBDepwDw1oAYMRPMMrvDnSeXHenY9la8HJZ07hj5zPH9NFjXs= 2 4 6 8 10 12 14 16 c zIR /zIR AAACBHicbZC7TsMwFIadcivlFmDsYlEhMZUEIcFYwQJbQfQitSFyXKe1ajuR7SCVKAMLr8LCAEKsPAQbb4PbZigtv2Tp83/OkX3+IGZUacf5sQpLyyura8X10sbm1vaOvbvXVFEiMWngiEWyHSBFGBWkoalmpB1LgnjASCsYXo7rrQciFY3EnR7FxOOoL2hIMdLG8u1y+uinXcnh9W2WHc9c7nHm2xWn6kwEF8HNoQJy1X37u9uLcMKJ0JghpTquE2svRVJTzEhW6iaKxAgPUZ90DArEifLSyRIZPDROD4aRNEdoOHFnJ1LElRrxwHRypAdqvjY2/6t1Eh2eeykVcaKJwNOHwoRBHcFxIrBHJcGajQwgLKn5K8QDJBHWJreSCcGdX3kRmidV1/DNaaV2kcdRBGVwAI6AC85ADVyBOmgADJ7AC3gD79az9Wp9WJ/T1oKVz+yDP7K+fgHEkpgn Figure 6. Masses of the two lightest singlet scalars, S1 and S2 , lightest adjoint scalar (a0 ), lightest axial-vector (a1 ) and Fπ , normalized to the vector mass for m̂2b = −1, λ = 1, λ2 = −2, g52 = 1 and κ̂2 = 1 (4) for the solid (dashed) line. The axial-vector spectrum depends directly on the φ profile via eq. (3.13), being this responsible for the mass splitting from the vector spectrum. Another important quantity is the Goldstone decay constant Fπ , that is the order parameter of the chiral breaking. This can be calculated via holography from the axial-vector two-point correlator at zero momentum [33–35]: M5 L ∂z A(z) Fπ = ΠA (0) = − , (3.40) 2zUV A(z) zUV where A(z) is the 5D solution of the axial-vector with Dirichlet UV-boundary condition. The results (with no approximations) are shown in figure 6 for some representative values of the parameter space. Following the notation in QCD, we denote with a0 and a1 the adjoint scalar and axial-vector respectively. Since Fπ is the only quantity that depends on M5 (Nc in the dual theory), we have fixed its value using eq. (4.4) with Nc = 3. For zIR ≈ zIR c (limit I) where the chiral breaking is small, we see that indeed F and π (mρ − ma1 )/mρ are small. As we increase zIR , we move towards limit II where the breaking of the chiral symmetry is larger, as can be appreciated by the growth of Fπ and S2 − a0 and ρ − a1 mass splittings. On the other hand, the mass of a0 strongly depends on λ2 , and we have chosen a negative value, λ2 = −2, that makes the mass splitting with the singlet sector positive, as lattice simulations (see later) seem to suggest. Similarly to the singlet scalars, we also have that ma0 goes to zero as zIR → zIR c , since the tachyon value goes to zero in this limit and we recover the chiral symmetry. Let us briefly comment on what happens for other values of the parameters of the model. The effect of κ̂2 in the mass spectrum is clear from figure 6 where we show the spectrum for two different values of κ̂2 . The main effect is that as we increase κ̂2 , the profile of φ becomes flatter and smaller, as appreciated in figure 2, giving a smaller breaking of the chiral symmetry. The spectrum is mildly sensitive to the values of m̂2b , unless we take – 18 – 1.6 1.6 1.4 a1 1.4 a1 AAAB6nicbZBNS8NAEIYn9avWr6hHL4tF8FQSEfRY9OKxov2ANpTJdtMu3WzC7kYooT/BiwdFvPqLvPlv3LY5aOsLCw/vzLAzb5gKro3nfTultfWNza3ydmVnd2//wD08aukkU5Q1aSIS1QlRM8ElaxpuBOukimEcCtYOx7ezevuJKc0T+WgmKQtiHEoecYrGWg/Y9/tu1at5c5FV8AuoQqFG3/3qDRKaxUwaKlDrru+lJshRGU4Fm1Z6mWYp0jEOWdeixJjpIJ+vOiVn1hmQKFH2SUPm7u+JHGOtJ3FoO2M0I71cm5n/1bqZia6DnMs0M0zSxUdRJohJyOxuMuCKUSMmFpAqbncldIQKqbHpVGwI/vLJq9C6qPmW7y+r9ZsijjKcwCmcgw9XUIc7aEATKAzhGV7hzRHOi/PufCxaS04xcwx/5Hz+AOiZjYk= 1.2 1.2 AAAB6nicbZBNS8NAEIYn9avWr6hHL4tF8FQSEfRY9OKxov2ANpTJdtMu3WzC7kYooT/BiwdFvPqLvPlv3LY5aOsLCw/vzLAzb5gKro3nfTultfWNza3ydmVnd2//wD08aukkU5Q1aSIS1QlRM8ElaxpuBOukimEcCtYOx7ezevuJKc0T+WgmKQtiHEoecYrGWg/Y9/tu1at5c5FV8AuoQqFG3/3qDRKaxUwaKlDrru+lJshRGU4Fm1Z6mWYp0jEOWdeixJjpIJ+vOiVn1hmQKFH2SUPm7u+JHGOtJ3FoO2M0I71cm5n/1bqZia6DnMs0M0zSxUdRJohJyOxuMuCKUSMmFpAqbncldIQKqbHpVGwI/vLJq9C6qPmW7y+r9ZsijjKcwCmcgw9XUIc7aEATKAzhGV7hzRHOi/PufCxaS04xcwx/5Hz+AOiZjYk= mi 1.0 1.0 m⇢ 0.8 a0 S2 AAAB6nicbZBNS8NAEIYn9avWr6pHL4tF8FSSIuix6MVjpfYD2lA22027dLMJuxOhhP4ELx4U8eov8ua/cdvmoK0vLDy8M8POvEEihUHX/XYKG5tb2zvF3dLe/sHhUfn4pG3iVDPeYrGMdTeghkuheAsFSt5NNKdRIHknmNzN650nro2I1SNOE+5HdKREKBhFazWbg9qgXHGr7kJkHbwcKpCrMSh/9YcxSyOukElqTM9zE/QzqlEwyWelfmp4QtmEjnjPoqIRN362WHVGLqwzJGGs7VNIFu7viYxGxkyjwHZGFMdmtTY3/6v1Ugxv/EyoJEWu2PKjMJUEYzK/mwyF5gzl1AJlWthdCRtTTRnadEo2BG/15HVo16qe5YerSv02j6MIZ3AOl+DBNdThHhrQAgYjeIZXeHOk8+K8Ox/L1oKTz5zCHzmfP9TJjXw= 0.8 a0 AAAB6nicbZBNS8NAEIYn9avWr6hHL4tF8FQSEfRY9OKxov2ANpTNdtMu3WzC7kQooT/BiwdFvPqLvPlv3LY5aOsLCw/vzLAzb5hKYdDzvp3S2vrG5lZ5u7Kzu7d/4B4etUySacabLJGJ7oTUcCkUb6JAyTup5jQOJW+H49tZvf3EtRGJesRJyoOYDpWIBKNorQfa9/pu1at5c5FV8AuoQqFG3/3qDRKWxVwhk9SYru+lGORUo2CSTyu9zPCUsjEd8q5FRWNugny+6pScWWdAokTbp5DM3d8TOY2NmcSh7YwpjsxybWb+V+tmGF0HuVBphlyxxUdRJgkmZHY3GQjNGcqJBcq0sLsSNqKaMrTpVGwI/vLJq9C6qPmW7y+r9ZsijjKcwCmcgw9XUIc7aEATGAzhGV7hzZHOi/PufCxaS04xcwx/5Hz+AOcVjYg= S2 AAAB6nicbZBNS8NAEIYn9avWr6pHL4tF8FSSIuix6MVjpfYD2lA22027dLMJuxOhhP4ELx4U8eov8ua/cdvmoK0vLDy8M8POvEEihUHX/XYKG5tb2zvF3dLe/sHhUfn4pG3iVDPeYrGMdTeghkuheAsFSt5NNKdRIHknmNzN650nro2I1SNOE+5HdKREKBhFazWbg9qgXHGr7kJkHbwcKpCrMSh/9YcxSyOukElqTM9zE/QzqlEwyWelfmp4QtmEjnjPoqIRN362WHVGLqwzJGGs7VNIFu7viYxGxkyjwHZGFMdmtTY3/6v1Ugxv/EyoJEWu2PKjMJUEYzK/mwyF5gzl1AJlWthdCRtTTRnadEo2BG/15HVo16qe5YerSv02j6MIZ3AOl+DBNdThHhrQAgYjeIZXeHOk8+K8Ox/L1oKTz5zCHzmfP9TJjXw= 0.6 0.6 S1 AAAB+3icbZDLSgMxFIbPeK31VuvSTbAIrsqMCLosunFZwV6gMwyZNNOGJpMhyYhlmFdx40IRt76IO9/GtJ2Fth5I+Pj/c8jJH6WcaeO6387a+sbm1nZlp7q7t39wWDuqd7XMFKEdIrlU/QhryllCO4YZTvupolhEnPaiye3M7z1SpZlMHsw0pYHAo4TFjGBjpbBW92OFSS5CVtjLV2NZhLWG23TnhVbBK6EBZbXD2pc/lCQTNDGEY60HnpuaIMfKMMJpUfUzTVNMJnhEBxYTLKgO8vnuBTqzyhDFUtmTGDRXf0/kWGg9FZHtFNiM9bI3E//zBpmJr4OcJWlmaEIWD8UZR0aiWRBoyBQlhk8tYKKY3RWRMbZhGBtX1YbgLX95FboXTc/y/WWjdVPGUYETOIVz8OAKWnAHbegAgSd4hld4cwrnxXl3Phata045cwx/yvn8Ac+3lO0= 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AAAB6nicbZBNS8NAEIYn9avWr6pHL4tF8FQSEeqx6MVjpfYD2lA22027dLMJuxOhhP4ELx4U8eov8ua/cdvmoK0vLDy8M8POvEEihUHX/XYKG5tb2zvF3dLe/sHhUfn4pG3iVDPeYrGMdTeghkuheAsFSt5NNKdRIHknmNzN650nro2I1SNOE+5HdKREKBhFazWbA29QrrhVdyGyDl4OFcjVGJS/+sOYpRFXyCQ1pue5CfoZ1SiY5LNSPzU8oWxCR7xnUdGIGz9brDojF9YZkjDW9ikkC/f3REYjY6ZRYDsjimOzWpub/9V6KYY3fiZUkiJXbPlRmEqCMZnfTYZCc4ZyaoEyLeyuhI2ppgxtOiUbgrd68jq0r6qe5YfrSv02j6MIZ3AOl+BBDepwDw1oAYMRPMMrvDnSeXHenY9la8HJZ07hj5zPH9NFjXs= 0.4 0.2 0.2 2 3 4 5 6 2 4 6 8 10 12 14 c c zIR /zIR zIR /zIR JHEP10(2019)202 AAACBHicbZC7TsMwFIadcivlFmDsYlEhMZUEIcFYwQJbQfQitSFyXKe1ajuR7SCVKAMLr8LCAEKsPAQbb4PbZigtv2Tp83/OkX3+IGZUacf5sQpLyyura8X10sbm1vaOvbvXVFEiMWngiEWyHSBFGBWkoalmpB1LgnjASCsYXo7rrQciFY3EnR7FxOOoL2hIMdLG8u1y+uinXcnh9W2WHc9c7nHm2xWn6kwEF8HNoQJy1X37u9uLcMKJ0JghpTquE2svRVJTzEhW6iaKxAgPUZ90DArEifLSyRIZPDROD4aRNEdoOHFnJ1LElRrxwHRypAdqvjY2/6t1Eh2eeykVcaKJwNOHwoRBHcFxIrBHJcGajQwgLKn5K8QDJBHWJreSCcGdX3kRmidV1/DNaaV2kcdRBGVwAI6AC85ADVyBOmgADJ7AC3gD79az9Wp9WJ/T1oKVz+yDP7K+fgHEkpgn AAACBHicbZC7TsMwFIadcivlFmDsYlEhMZUEIcFYwQJbQfQitSFyXKe1ajuR7SCVKAMLr8LCAEKsPAQbb4PbZigtv2Tp83/OkX3+IGZUacf5sQpLyyura8X10sbm1vaOvbvXVFEiMWngiEWyHSBFGBWkoalmpB1LgnjASCsYXo7rrQciFY3EnR7FxOOoL2hIMdLG8u1y+uinXcnh9W2WHc9c7nHm2xWn6kwEF8HNoQJy1X37u9uLcMKJ0JghpTquE2svRVJTzEhW6iaKxAgPUZ90DArEifLSyRIZPDROD4aRNEdoOHFnJ1LElRrxwHRypAdqvjY2/6t1Eh2eeykVcaKJwNOHwoRBHcFxIrBHJcGajQwgLKn5K8QDJBHWJreSCcGdX3kRmidV1/DNaaV2kcdRBGVwAI6AC85ADVyBOmgADJ7AC3gD79az9Wp9WJ/T1oKVz+yDP7K+fgHEkpgn Figure 7. Masses of the two lightest singlet scalars, S1 and S2 , lightest adjoint scalar (a0 ), lightest axial-vector (a1 ) normalized to the lightest vector mass (mρ ) for constant Fπ ∼ mρ /7 as a function c of zIR /zIR for λ = 1. We have taken κ̂2 = 1 (4) for solid (dashed) lines. The left grey (right orange) band corresponds to the region 0.5 < g52 < 2 for κ̂2 = 1 (4). LEFT: m̂2b = −1. RIGHT: m̂2b = −1.5. m̂2b < −2 that we will discuss later (figure 8). Finally, g52 only affects Fπ and ma1 that will increase as g52 increases. It is more instructive, also in part to compare later our results with lattice simulations, to analyze the spectrum at constant Fπ . For this purpose, we adjust g52 to fulfill, for the different values of zIR /zIR c (or equivalently δ Λ̂), the relation F ' m /7 as in QCD. The π ρ results are given in figure 7 for m̂2b = −1 (left) and m̂2b = −1.5 (right). We have kept g52 in the interval 0.5 < g52 < 2 and this has limited the possible values of zIR /zIRc to the blue and orange bands for κ̂2 = 1 and 4 respectively.12 The main conclusions from figure 7 are the following. The lightest resonance is always the scalar S1 , a dilaton-like state. The S2 , the Higgs-like state, is also smaller or around mρ , and can only be larger if we take large values of zIR /zIR c (that implies small values of g 2 in order to keep F ∼ m /7). The ratio 5 π ρ ma1 /mρ is closer to 1 than in real QCD where ma1 /mρ ∼ 1.6 or previous holographic QCD versions [33–35]. The mass of a0 is also smaller than in real QCD. As we will see in the following, these properties are also found in lattice QCD for large NF . The situation is only slightly modified in the region m̂2b < −2. In figure 8 we show the mass spectrum for m̂2b = −3. The main differences with respect figure 7 is in the scalar mass spectrum where we appreciate that at smaller values of zIR /zIR c 0 , where here √ c 0 zIR ≡ e π/  zUV , the a0 and S2 masses do not go to zero. The reason is the following. As explained in section 3.2.2, for m̂2b < −2 the profile of φ is always non-zero (unless zIR < zIRc ∼ z UV ). This implies that we do not recover the chiral symmetry in the region of interest, zIR ∼ zIR c 0 , and the S and a masses never approach zero. Nevertheless, their 2 0 masses are predicted to be around the ρ mass. The main lesson that we have learned on the mass spectrum of S2 , a0 and a1 is that they seem to tend to be lighter in models close to the conformal transition (as compared to real QCD that is far from the conformal critical point). What is the reason for that? c 12 The constraint δ Λ̂ ≤ 0 has not been imposed. If we impose it, we obtain zIR /zIR ≥ 3.06 (3.37) for c 2 κ̂ = 1 (4) in the left plot of figure 7, and zIR /zIR ≥ 1.88 (1.9) for the right plot. – 19 – 1.6 1.4 a1 AAAB6nicbZBNS8NAEIYn9avWr6hHL4tF8FQSEfRY9OKxov2ANpTJdtMu3WzC7kYooT/BiwdFvPqLvPlv3LY5aOsLCw/vzLAzb5gKro3nfTultfWNza3ydmVnd2//wD08aukkU5Q1aSIS1QlRM8ElaxpuBOukimEcCtYOx7ezevuJKc0T+WgmKQtiHEoecYrGWg/Y9/tu1at5c5FV8AuoQqFG3/3qDRKaxUwaKlDrru+lJshRGU4Fm1Z6mWYp0jEOWdeixJjpIJ+vOiVn1hmQKFH2SUPm7u+JHGOtJ3FoO2M0I71cm5n/1bqZia6DnMs0M0zSxUdRJohJyOxuMuCKUSMmFpAqbncldIQKqbHpVGwI/vLJq9C6qPmW7y+r9ZsijjKcwCmcgw9XUIc7aEATKAzhGV7hzRHOi/PufCxaS04xcwx/5Hz+AOiZjYk= 1.2 mi 1.0 S2 m⇢ 0.8 a0 AAAB6nicbZBNS8NAEIYn9avWr6pHL4tF8FSSIuix6MVjpfYD2lA22027dLMJuxOhhP4ELx4U8eov8ua/cdvmoK0vLDy8M8POvEEihUHX/XYKG5tb2zvF3dLe/sHhUfn4pG3iVDPeYrGMdTeghkuheAsFSt5NNKdRIHknmNzN650nro2I1SNOE+5HdKREKBhFazWbg9qgXHGr7kJkHbwcKpCrMSh/9YcxSyOukElqTM9zE/QzqlEwyWelfmp4QtmEjnjPoqIRN362WHVGLqwzJGGs7VNIFu7viYxGxkyjwHZGFMdmtTY3/6v1Ugxv/EyoJEWu2PKjMJUEYzK/mwyF5gzl1AJlWthdCRtTTRnadEo2BG/15HVo16qe5YerSv02j6MIZ3AOl+DBNdThHhrQAgYjeIZXeHOk8+K8Ox/L1oKTz5zCHzmfP9TJjXw= AAAB6nicbZBNS8NAEIYn9avWr6hHL4tF8FQSEfRY9OKxov2ANpTNdtMu3WzC7kQooT/BiwdFvPqLvPlv3LY5aOsLCw/vzLAzb5hKYdDzvp3S2vrG5lZ5u7Kzu7d/4B4etUySacabLJGJ7oTUcCkUb6JAyTup5jQOJW+H49tZvf3EtRGJesRJyoOYDpWIBKNorQfa9/pu1at5c5FV8AuoQqFG3/3qDRKWxVwhk9SYru+lGORUo2CSTyu9zPCUsjEd8q5FRWNugny+6pScWWdAokTbp5DM3d8TOY2NmcSh7YwpjsxybWb+V+tmGF0HuVBphlyxxUdRJgkmZHY3GQjNGcqJBcq0sLsSNqKaMrTpVGwI/vLJq9C6qPmW7y+r9ZsijjKcwCmcgw9XUIc7aEATGAzhGV7hzZHOi/PufCxaS04xcwx/5Hz+AOcVjYg= AAAB+3icbZDLSgMxFIbPeK31VuvSTbAIrsqMCLosunFZwV6gMwyZNNOGJpMhyYhlmFdx40IRt76IO9/GtJ2Fth5I+Pj/c8jJH6WcaeO6387a+sbm1nZlp7q7t39wWDuqd7XMFKEdIrlU/QhryllCO4YZTvupolhEnPaiye3M7z1SpZlMHsw0pYHAo4TFjGBjpbBW92OFSS5CVtjLV2NZhLWG23TnhVbBK6EBZbXD2pc/lCQTNDGEY60HnpuaIMfKMMJpUfUzTVNMJnhEBxYTLKgO8vnuBTqzyhDFUtmTGDRXf0/kWGg9FZHtFNiM9bI3E//zBpmJr4OcJWlmaEIWD8UZR0aiWRBoyBQlhk8tYKKY3RWRMbZhGBtX1YbgLX95FboXTc/y/WWjdVPGUYETOIVz8OAKWnAHbegAgSd4hld4cwrnxXl3Phata045cwx/yvn8Ac+3lO0= 0.6 0.4 S1 0.2 AAAB6nicbZBNS8NAEIYn9avWr6pHL4tF8FQSEeqx6MVjpfYD2lA22027dLMJuxOhhP4ELx4U8eov8ua/cdvmoK0vLDy8M8POvEEihUHX/XYKG5tb2zvF3dLe/sHhUfn4pG3iVDPeYrGMdTeghkuheAsFSt5NNKdRIHknmNzN650nro2I1SNOE+5HdKREKBhFazWbA29QrrhVdyGyDl4OFcjVGJS/+sOYpRFXyCQ1pue5CfoZ1SiY5LNSPzU8oWxCR7xnUdGIGz9brDojF9YZkjDW9ikkC/f3REYjY6ZRYDsjimOzWpub/9V6KYY3fiZUkiJXbPlRmEqCMZnfTYZCc4ZyaoEyLeyuhI2ppgxtOiUbgrd68jq0r6qe5YfrSv02j6MIZ3AOl+BBDepwDw1oAYMRPMMrvDnSeXHenY9la8HJZ07hj5zPH9NFjXs= 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 JHEP10(2019)202 c0 zIR /zIR AAACDXicbZDLSgMxFIYz9VbrbdSlm2AVXEidEUGXRTe6q2Iv0BmHTJq2oUlmSDJCHeYF3Pgqblwo4ta9O9/GtB1Qqz8EPv5zDifnD2NGlXacT6swMzs3v1BcLC0tr6yu2esbDRUlEpM6jlgkWyFShFFB6ppqRlqxJIiHjDTDwdmo3rwlUtFIXOthTHyOeoJ2KUbaWIG9k94FqSc5vLjKsoNvvkmxtw+9WFJOssAuOxVnLPgX3BzKIFctsD+8ToQTToTGDCnVdp1Y+ymSmmJGspKXKBIjPEA90jYoECfKT8fXZHDXOB3YjaR5QsOx+3MiRVypIQ9NJ0e6r6ZrI/O/WjvR3RM/pSJONBF4sqibMKgjOIoGdqgkWLOhAYQlNX+FuI8kwtoEWDIhuNMn/4XGYcU1fHlUrp7mcRTBFtgGe8AFx6AKzkEN1AEG9+ARPIMX68F6sl6tt0lrwcpnNsEvWe9fOKGbog== Figure 8. As in figure 7 but for m̂2b = −3. As it is well-known, the dimension of a scalar operator has a minimal value determined by its unitarity bound, in this case Dim[O∗ ] = 1, a limit at which the scalar decouples from the CFT [40]. Therefore it is expected that, as a scalar operator approaches this decoupling limit, the mass of the lightest resonance associated to it becomes smaller. By using the AdS/CFT correspondence this means, via eq. (2.4), that the lightest Φ resonance is expected to be lighter the more we approach the BF-bound. We can also √ understand 2+ 4+MΦ 2 L2 this “geometrically”. The wave-function of the lightest scalar grows as z , that implies that the wave-function becomes flatter and spread more into the AdS 5 space as we approach the BF-bound MΦ2 L2 → −4. In this limit, then, the scalar excitation becomes less sensitive to the IR and therefore its mass is expected to be smaller (see also [41]).13 The fact that the profile of φ becomes flatter as we approach the BF-bound also explains the smaller mass splitting between a1 and ρ than in real QCD. Indeed, if we keep Fπ constant, the flatter the φ profile, the smaller φ(zIR ). Since the a1 wave-function is peaked towards the IR-brane, it is mostly sensitive to the value of φ(zIR ). Therefore, as the 5D mass of φ gets closer to the BF-bound, we expect ma1 to be less sensitive to chiral breaking. For the same reason we understand ma1 becoming smaller as we increase zIR (see figure 7), as φ becomes flatter for larger zIR . 4 Comparison with lattice QCD in the large NF Lattice results for Nc = 3 QCD with NF = 8 have been reported in refs. [1–5]. At such large value of NF , it is believed that QCD is close to the conformal transition, expected to 13 We could make the wave-function even flatter by quantizing differently p the scalar following ref. [39] (valid for −4 < (MΦ L)2 < −3). In this case we have Dim[O∗ ] = 2 − 4 + MΦ2 L2 that means that the scalar √ is the dual of an operator of dimension between 1 and 2. We will get in this case the wave-function 2− 4+MΦ 2 L2 z , reaching the full decoupling from the IR (from the CFT) at MΦ2 = −3/L2 when the mode becomes non-normalizable. Nevertheless the scalar becomes UV sensitive and it is not expected to survive in the spectrum. – 20 – occur around NF ∼ 9. It was found [1–5] Fπ mf0 ma 0 ma1 ' 0.14 , ' 0.5 , ' 1, ' 1.4 , (4.1) mρ mρ mρ mρ where f0 is the lightest flavor-singlet 0++ state (S1 in our notation).14 It is instructive to compare them with real QCD that is supposed to be far from the conformal edge. We have [42] Fπ mf0 ma0 ma 1 ' 0.13 , ' 1.3 , ' 1.3 , ' 1.6 . (4.2) mρ mρ mρ mρ JHEP10(2019)202 We see that close to the conformal transition, we spectrum of eq. (4.1) shows, as compared to real QCD eq. (4.2), lighter f0 and a0 scalars, and a smaller mass splitting between the ρ and a1 resonance. Surprisingly, the ratio of Fπ /mρ is quite similar to real QCD, showing that this quantity is quite independent of NF . Let us compare our results to the values of eq. (4.1). In order to reduce the number of parameters, we can match the predictions of our model at the UV with those of QCD with NF flavors. In particular, the two-point vector-vector correlator at large momentum p2 is given in our model by [34, 35] M5 L 2 ΠV (p2 ) ' − p ln(p2 zUV 2 ), (4.3) 2g52 that matching to that of QCD with NF flavors gives M5 L Nc 2 = . (4.4) g5 12π 2 Using eq. (4.4) our predictions for the mass spectrum were presented in figures 7 and 8 for Fπ = mρ /7. We see that our predictions on the spectrum of resonances follow quite close the pattern eq. (4.1). We have ma1 /mρ closer to one than in QCD, with the scalars a0 and S1 being lighter than the ρ in most of the parameter space. Indeed, in the region 1.2 < ma1 /mρ < 1.4, we find ma0 /mρ . 1 and mS1 /mρ . 0.3. There are other important predictions arising from our holographic model that would be interesting to check in future lattice simulations. For example, as we already mentioned, the mass splittings between the adjoint and singlet vectors is zero at leading order, and can only arise from loop effects or higher-dimensional operators that are suppressed. Also the second singlet scalar S2 (a Higgs-like scalar) seems to be lighter than the ρ in the region where 1.2 < ma1 /mρ < 1.4. Finding this second resonance so light would be a clear indication that the lightest scalar S1 is a dilaton and not a Higgs-like state. Other properties of the scalars, such as decay constants or couplings, that can also be calculated in these holographic models, are left for future work. 14 Lattice results are presented for nonzero quark masses. We will assume here that the pattern eq. (4.1) does not drastically change in the limit Mq → 0. – 21 – 5 Models for the hierarchy problem The model described here open new possibilities for generating small scales. Since the c ), we have a way to generate exponentially IR-brane is naturally stabilized at zIR ∼ O(zIR small scales. Indeed, from eq. (3.21) we have 1 1 √ 1 1 ∼ c = e−π/   . (5.1) zIR zIR zUV zUV JHEP10(2019)202 The presence of a scalar with a mass just below the BF bound can also be achieved dynamically. If the mass of Φ is z-dependent, for example, MΦ2 L2 = −4 − E(z), where E(z) slowly varies from negative to positive values as z increases, the mass of Φ will cross the BF bound at the position z = zUV0 at which E(zUV0 ) = 0. For example, we can consider E(z) = 0 ) with   1 (for other cases, see [43]). This z-dependent mass for Φ can be easily  ln(z/zUV √ achieved by promoting E(z) to a scalar R with a 5D potential V = − R(1 + |Φ|2 /L2 ). √ This scalar gets a profile R(z) =  ln(z/zUV ), giving a contribution to the mass of Φ proportional to  ln(z/zUV ). 0 ), the wave-function of the massless For MΦ2 L2 = −4 − E(z) with E(z) =  ln(z/zUV mode is not anymore eq. (3.20) but 2 √ 3/2 z   φt (x) 2 p φ(z) = z ln(z/zUV ) J1/3  ln , (5.2) N 3 zUV where J1/3 is a Bessel-function of order 1/3, and the IR-boundary condition eq. (3.17) at c leads now in the limit  → 0 to zIR = zIR c 2 √ 3/2 zIR   1  ln ' n− π , n = 1, 2, . . . , (5.3) 3 zUV 12 corresponding to the zeros of the Bessel function. The situation is quite similar to the case of constant E(z) discussed above; the only important difference worth to mention is that in the limiting case II with λ > 0, the maximal value of φ is not constant, as MΦ2 evolves logarithmically. The theory has evolved into a deformed CFT. We leave the implications of these scenarios for the electroweak scale for future work. We only point out several interesting features. First, the lightness of the dilaton can have important implications for the LHC [44–46]. Also the fact that the mass of a1 is closer to the mass of ρ implies smaller values for the S-parameter (this was also pointed out in ref. [47] for MΦ2 → −4/L2 from above), favored by precision experimental data. Furthermore, having the operator that drives symmetry breaking a dimension close to 2 helps to pass flavor constraints [48]. Also it was shown in ref. [49] that these models can lead to a long period of supercooling in the early universe with implications in Dark Matter and axion cosmological abundances. – 22 – 6 Conclusions We have used holography to study strongly-coupled theories close to the conformal tran- sition, that is the transition from the non-conformal regime to the conformal one. This transition is expected to happen in gauge theories (such as QCD) as the number of fermions NF increases. Recent lattice results [1–5] have shown that as we get closer to the conformal transition, the lightest resonance is a 0++ state, claimed to be a dilaton. We have followed the idea of ref. [6] that suggested that conformality is lost when the IR fixed point merges with a UV fixed point, as shown in figure 1. Holography tells that this must occur by an operator O∗ (probably q q̄ in QCD) whose dimension is equal to two JHEP10(2019)202 that gets a small imaginary part when leaving the conformal regime. In the gravitational dual models this is driven by a scalar whose mass goes below the BF bound and becomes tachyonic. We have presented a very simple extra-dimensional model with the essential ingredients to study the conformal transition and calculate the mass spectrum. The model consists of a five-dimensional gravitational sector with a scalar and gauge bosons associated to the global SU(NF )L ⊗ SU(NF )R ⊗ U(1)B . We have allowed for the most general Lagrangian following the 5D EFT rules, and explained the connection between the 5D couplings and the large Nc and NF expansion. To model confinement we cut off the space by an IR-brane that we showed to be stabilized by the presence of the tachyon. We have calculated the mass spectrum of this 5D model, showing that indeed the dila- ton corresponds to the lightest resonance. To understand this property, we have derived a simple formula for the dilaton mass, eq. (3.32). This shows that the mass of the dilaton crucially depends on βφ (zIR ) given in eq. (3.35) that is sensitive to the variation of φ(zIR ) as we move the UV boundary (therefore sensitive to the explicit breaking of the conformal symmetry). Either for small or large values of zIR , we have shown that βφ (zIR ) → 0 and therefore the dilaton mass tends to zero. For small zIR this is due to either the existence of an inflection point in the dilaton potential (for m̂2b > −2), or that φ(zIR ) becomes constant (for m̂2b < −2). Also for small φ(z), where we can perform analytical calculations, we find a mild log-dependence of φ(zIR ) with zIR (and therefore a smallish βφ (zIR )) that can be traced back to the explicit breaking of the conformal symmetry due to the double-trace marginal operator Og = |O∗ |2 . For large zIR , also βφ (zIR ) → 0 as the tachyon either goes to the min- imum of its potential and becomes constant or enter into a “slow-roll” condition, meaning that the geometry approaches again AdS5 . In between these two limiting cases, the dilaton can become heavier but its mass cannot grow enough to overcome mρ . Therefore the dila- ton is found to be lighter than the rest of the resonances, although it is never parametrically lighter in most of the area of the allowed parameter space, as shown in figure 3. We have compared our predictions with lattice results for QCD with a large NF (eq. (4.1)) and showed that our model predicts quite similar resonance mass pattern: the lightest state is the singlet 0++ , with the adjoint scalar a0 mass close to mρ and lighter than in real QCD. We have also shown than the mass splitting between the vector (ρ) and axial-vector (a1 ) is smaller close to the conformal transition. We have given a geometric explanation for these properties. Furthermore, the 5D model proposed here also provides – 23 – extra predictions that lattice could check in the future. For example, we find that the sec- ond 0++ state, S2 , is mostly a Higgs-like state (q q̄ state) with a mass around mρ , similarly as a0 . The 5D model also predicts that the masses of the flavor singlet and adjoint vector resonances are similar (as it happens also in real QCD). There are several interesting calculations that are left for the future. For example, it is also possible to calculate decay constants and couplings of the resonances along the lines of refs. [34, 35]. One could also easily add explicit quark masses to the model to see the impact on the spectrum, or study the model at the conformal edge but inside the conformal window. It could also be interesting to understand what are the holographic JHEP10(2019)202 versions of the complex CFT described in ref. [30]. Finally, as discussed above, this type of models can provide a new approach to the hierarchy problem with a clear impact on LHC phenomenology as the 0++ resonance is expected to be the lightest one. All these issues clearly deserve more attention. Acknowledgments We would like to thank Eugenio Megias, Giuliano Panico and Mariano Quiros. AP was supported by the Catalan ICREA Academia Program. LS was supported by a Beca Pre- doctoral Severo Ochoa del Ministerio de Economı́a y Competitividad (SVP-2014-068850). This work was also partly supported by the grants FPA2017-88915-P, 2017-SGR-1069 and SEV-2016-0588. A Scalar and gravity coupled equations of motion In this appendix we present the equations of motion (EOM) of the scalar and gravitational sector, that we use in this article in order to derive the background and mass spectrum of the model. For this purpose it is useful to work with proper coordinates, as the metric- scalar system of EOM simplifies. Once the results are obtained, we have rewritten them in conformal coordinates eq. (3.5) to be presented in the main text. Conformal coordinates allow a better interpretation of the results as 1/z determines the natural mass scale at the position z. A.1 Scalar-metric system In proper coordinates {xµ , y} the background metric can be written as ds2 = e−2A(y) ηµν dxµ dxν − dy 2 , (A.1) where ηµν = diag(1, −1, −1, −1), 0 ≤ y ≤ yIR with the IR-brane localized at y = yIR , and we have conveniently rewritten the warp factor as a = e−A . The 5D EOM for the metric-scalar system, that follow from the action in eq. (3.1) in these coordinates, are – 24 – given by φ̈ = 4Ȧφ̇ + V 0 , (A.2) s 1 κ̂2 L2  φ̇2  Ȧ = + − V (φ) , (A.3) L2 12 2 κ̂2 L2 2 Ä = φ̇ , (A.4) 6 where in this appendix φ̇ ≡ ∂y φ, Ȧ ≡ ∂y A. At the IR-brane we must impose the IR- boundary conditions: JHEP10(2019)202   M5 φ̇ + Vb0 (φ) = 0, (A.5) yIR   6M5 Ȧ + V b (φ) = 0, (A.6) κ̂2 L2 yIR where the second equation is the junction condition that determines the value yIR where the IR-brane is dynamically stabilized. Plugging eq. (A.3) into eq. (A.2) gives a differential equation involving only φ that can be easily solved. Afterwards, we can solve eq. (A.3) to obtain the metric warp factor A(y). We can go to conformal coordinates by using dy/dz = e−A(y) . Working with proper coordinates, the slow-roll conditions are, in analogy with inflation, given by h i Ḣ Ḣ 1 ∂ y H2  1 and h i  1, (A.7) H2 H Ḣ H2 where H = Ȧ. Using eq. (A.2)–eq. (A.4), the slow-roll conditions eq. (A.7) can be written in the following equivalent form: (V 0 )2 2 2 V 00 2 2 12 2  κ̂ L and 12  κ̂ L . (A.8) (V − κ̂2 L4 ) V − κ̂2 L4 Since we work with polynomial potentials, the two slow-roll conditions eq. (A.8) reduce to one condition when the feedback of the metric becomes important (V  κ̂212L4 ). This is given by κ̂2 φ2  1/L2 . (A.9) A.2 Singlet scalar-dilaton system If the dilaton is not a priori assumed to be light, we must solve exactly the eigenmasses of the scalar sector considering the mixing between the singlet scalar Φs and the dilaton, which is of order one for NF ∼ Nc . This is done conveniently in a diagonal gauge where the brane is straight, corresponding to a constant value of the extra coordinate yIR =const. In this gauge, the EOM reduces to eq. (3.17) of ref. [37], that we can write as 3m2n 2A 3e2A   −2A Dψn = e ψn with D ≡ 1 − ∂y ∂y [e ] , (A.10) κ̂2 φ̇2 L2 κ̂2 φ̇2 L2 – 25 – with the IR-boundary condition: ! φ̈ m2n ψn = Vb00 + ∂y [e−2A ψn ] . (A.11) yIR φ̇ yIR A.2.1 Light dilaton limit When the dilaton becomes the lightest mode of the scalar sector, we can analytically derive its mass, as given in eq. (3.31). Here we present the details to obtain this mass. The physical meaning of the dilaton field is the IR scale that appears dynamically in the theory. In the 5D model, this is incarnated geometrically by the location of the IR-brane JHEP10(2019)202 which is indeed dynamical. The picture is more transparent by allowing the IR position to be xµ -dependent, i.e., that it is a 4D field. This is equivalent to using a gauge where the IR-brane location is not straight, but rather defined by the surface y = yIR (xµ ). On the other hand, the variable that transforms under a scale dilatation of the xµ coordinates is not the proper coordinate y itself but the warp factor a(y). Therefore it is natural to identify the dilaton field with the warp factor evaluated at the IR-brane location, 1 −A(yIR ) φ̂d = e . (A.12) L This variable is also convenient because it is easy to extract the normalization of both the potential and of the kinetic terms inpterms of it. For instance, with this definition the brane tension term (proportional to −g IR with gµν IR the induced metric on the brane) is simply a quartic coupling, φ̂4d . Below we will see that this variable is actually not canonically normalized in general, although its kinetic term can be easily found. From the normalization of both kinetic and potential terms then a formula for the mass will follow. Let us start by the potential term. We can redo the argument around eq. (3.30) in terms of φ̂d , that allows to reconstruct quite directly the derivative of the off-shell effective potential, dVeff /dφ̂d , which must be proportional to φ̂d3 and to the junction condition. The overall normalization constant can be fixed by requiring that in the κ̂2 → 0 limit the effective potential reduces to Veff = (M5 L) bulk (V + φ̇2 /2) + Vb , with φ solving the EOM R and depending parametrically on yIR . This leads to s  02 2 3 dVeff (φ̂d ) M5 2 4 κ̂ L Vb  κ̂ L  3 = 24 2 L φ̂d  1 + κ 12 2M 2 −V + 6M5 Vb  , (A.13) dφ̂d 5 yIR that differs from eq. (3.30) by an overall multiplicative constant; this does not matter much however for the mass eq. (3.31) as long as we factor out the same constant in the kinetic term. Next, the normalization of the dilaton kinetic term. Another advantage of using a non-straight gauge is that all the kinetic term contributions arise only from localized terms on the IR-brane itself. This is welcome because the dilaton is an IR mode and its properties should arise from the IR only. Moreover, it is also convenient because it allows to identify these kinetic term contributions in the ‘probe’ limit, where we ignore how the brane bending sources the 5D metric. Following [50], one quickly sees that the radion/dilaton kinetic term – 26 – arises from two sources. First, the brane tension (potential) term, via the determinant of the induced metric on the brane, s p (∂yIR )2 −g IR = a4 (yIR ) 1 − 2 , (A.14) a (yIR ) where (∂yIR )2 = η µν ∂µ yIR ∂ν yIR . Second, the Gibbons-Hawking, proportional to the ex- trinsic curvature at the y = yIR (xµ ) surface, generates additional terms. The relevant ones (contributing to the quadratic kinetic part) are proportional to the derivative of the warp factor at the brane location. JHEP10(2019)202 At this point we must make a slight detour, to be more precise on how several quantities depend on yIR , that is, on the dilaton. The key point is that the bulk scalar φ is coupled to the IR-brane (because the IR potential Vb (φ) acts effectively like a scalar charge). For this reason, the profile of φ (and therefore of the metric) in the bulk actually depends on the IR-brane location even when we allow the brane location to be off shell. To make this dependence manifest, we can write that the field profile is a function of both the bulk coordinate and the IR-brane location, φ = φ(y, yIR ).15 This is indeed implied by eq. (3.20) and eq. (3.24) in the main text. In this notation, the field evaluated on the IR-brane is φ(yIR , yIR ), and the derivative with respect to yIR originates from the two arguments. The boundary condition specifies ∂y φ(y, yIR )|y=yIR , but the ‘full’ derivative ∂yIR φ(yIR , yIR ) is left unspecified and it is nontrivial in a nonlinear theory. As we will see shortly this full derivative is the one that controls the dilaton mass (it is the one that appears in eq. (3.31)). The same qualifications apply also for the metric. After all, the ‘Friedman’ eq. (A.3) forces ∂y a to be an algebraic function of φ(y, yIR ) and ∂y φ(y, yIR ), so the warp factor profile too depends parametrically on yIR , that is, we must write a = a(y, yIR ). Now, the extrinsic curvature is related to ∂y a(y, yIR )|y=yIR . The dilaton variable φ̂d defined in eq. (A.12) stands for a(yIR , yIR ), and its derivative with respect to yIR , ∂yIR a(yIR ) is not the same as ∂y a(y, yIR )|y=yIR . To make the distinction clear in the following we will keep this notation and show explicitly the difference. Form eq. (A.3) we have v ! Z yIR u u 1 2 L2 2 (ȳ, y ) κ̂ φ̇ IR − ln a(yIR , yIR ) = A(yIR , yIR ) = dȳ t 2 + − V (φ(ȳ, yIR )) , yUV L 12 2 (A.15) that differentiating with respect to yIR leads to κ̂2 L3 yIR φ̇ ∂y φ̇ − V 0 (φ) ∂yIR φ Z ∂yIR A(yIR , yIR ) = Ȧ(yIR , yIR ) + dȳ q IR , (A.16) 24 yUV 2 4 1 + κ̂ 12L (φ̇2 /2 − V (φ)) where Ȧ(yIR , yIR ) ≡ (∂y ln a(y, yIR )) y=yIR and the rest of the notation should be clear. This shows that both at small and large κ̂2 , the difference between Ȧ and ∂yIR A is small. 15 We could also include the dependence on the UV brane location, yUV , however we will omit it here to avoid clutter. The symmetries of the background ensure that φ = φ(y − yUV , yIR − yUV ). This makes manifest that the field evaluated at the IR brane, φ(yIR − yUV , yIR − yUV ), is sensitive to yUV . – 27 – Numerically, in our solutions we finds it to be less than 5%. (The same cannot be said about the two types of derivatives acting on φ.) Returning to the kinetic term: after collecting all terms and using the EOM of the background, one arrives at [50] M5 3 Ȧ(yIR ) a2 (yIR ) (∂µ yIR )2 . (A.17) κ2 With the definition eq. (A.12) that implies ∂µ φ̂d = −∂yIR ln a(yIR ) φ̂d ∂µ yIR , this gives 3M5 L2 Ȧ(yIR ) (∂µ φ̂d )2 . (A.18) κ2 [∂yIR A(yIR )]2 JHEP10(2019)202 As discussed above, one can set Ȧ(yIR ) ' ∂yIR A(yIR ) to a good approximation, therefore the kinetic term is to a good accuracy 3M5 L2 (∂µ φ̂d )2 . (A.19) κ2 ∂yIR A(yIR ) On the other hand, differentiating eq. (A.13) we get   0 00 d2 Veff (φ̂d )   4 2 1 Vb Vb 0 4 0 ∂yIR φ(yIR ) = −NF L M5 φ̂d 2 −V + Vb , (A.20) dφ̂2d Ȧ M5 M5 ∂yIR A(yIR ) by using the chain rule and eq. (A.12). Taking everything together, and the approximate expression eq. (A.19) we find that the physical dilaton mass is given by κ̂2 L4 2 1 Vb0 Vb00     2 0 4 0 mφd ' − φ̂d −V + V ∂yIR φ(yIR ) , (A.21) 6 Ȧ M52 M5 b where everything is evaluated at the minimum, which coincides with eq. (3.31) after the change of coordinates dy = a(z)dz. As a further cross-check, let us note that this expression agrees with eq. (2.25) of [51]. This was obtained starting directly from eq. (A.10) and obtaining an expression for the lowest KK mass under the assumption that it is light. The formula of [51] has two distinct limits corresponding to whether or not the light dilaton is incarnated by the IR-brane position. It is easy to check that in the limit where the dilaton is the displacement of the IR-brane, eq. (A.21) agrees with eq. (2.25) of [51]. Finally, we remark that using the EOM of the background we can rewrite eq. (A.17) as 1 − Vb φ(yIR ) a2 (yIR ) (∂µ yIR )2 .  (A.22) 2 One immediately realizes an important implication: the positivity of the kinetic energy restricts Vb (φ(yIR )) < 0. This is equivalent to restricting the effective tension on the IR- brane to be negative — as it should be in order that it gives an end to the geometry at yIR . If we demand that this constraint is satisfied by all the solutions, including the one with φ(z) = 0 (for  < 0), then this translates into a constraint on the IR-brane tension ‘detuning parameter’, δ Λ̂ < 6 . (A.23) This reproduces the upper bound in (3.29), i.e., that the brane on the IR (on the ‘interior’) side of the geometry has negative tension. – 28 – B A tale of two scalars: the 4D effective potential of a tachyon and a dilaton When both 4D tachyon and dilaton masses are smaller than 1/zIR , we can easily understand the physics of the system by just looking at the 4D effective theory for these two modes. This is possible when working close to the critical point, zIR ≈ zIRc (limiting case I of section 3.2), where we can obtain the masses and quartic couplings of the model as a perturbation of the model around zIR = zIRc . We find 1 1 1 1 Veff (φt , φd ) = m̂2 (φd )φ2t φ2d + λt φ4t + λd φ4d , (B.1) JHEP10(2019)202 M5 LNF 2 4 4 where φd = 1/zIR is the dilaton and φt the 4D tachyon. We are working here with non- canonically normalized fields: 1 1 3 Lkin = (∂µ φt )2 + 2 (∂µ φd )2 . (B.2) M5 LNF 2 κ̂ c such that m̂2 (φ ) is Eq. (B.1) can only give a non-trivial minimum for φd ≤ µc ≡ 1/zIR d negative. In this regime, this is given by m̂2 (φd ) = β ln(φd /µc ) , (B.3) where β is given in eq. (3.22). Recall that eq. (B.1) is only valid for m̂2 (φd )  1 that requires either | ln φd /µc |  1 or β  1 (m̂2b → −2). The tachyon quartic λt has a mild dependence on m̂2b and is derived below (section B.1), while the dilaton quartic is given by the detuning of the IR-brane tension: δ Λ̂ λd = 4 . (B.4) κ̂2 The presence of the tachyon leads to a non-trivial potential for the dilaton with a minimum at hφd i 1h p i ln = − 1 + 1 + 16λt λd /β 2 . (B.5) µc 4 We see that the largest value of the dilaton is given by ln hφd i/µc ≤ −1/4 that implies that c where the tachyon the IR-brane can never be stabilized very close to the critical point zIR mass is small. At the closest value ln hφd i/µc = −1/4, one finds that the minimum is an inflection point where the minimum coincides with a maximum (and one can check that the dilaton mass is zero at this point). Nevertheless, demanding that the quartic couplings are positive, to guarantee that for  < 0 the theory is conformal (φd → 0), one obtains ln hφd i/µc ≤ −1/2. We can also find the masses of the dilaton and tachyon by calculating the eigenvalues of the matrix of second derivatives at the minimum eq. (B.5) after canonically normalizing the fields. These give rather complicated functions of λt , λd and β. For β 2 /(λd λt )  1 they reduce to 2β κ̂2  2  2 2 2 κ̂ p mφ d ' p hφd i , m φt ' λd + λt /λd hφd i2 , (B.6) κ̂2 + 6 λt /λd 6 – 29 – 0.6 0.5 0.4 0.3 0.2 0.1 0.0 1 2 3 4 5 Figure 9. Masses of the two scalars, that play the role of S1 and S2 , as a function of µc /hφd i and JHEP10(2019)202 normalized to (3π/4) hφd i. We have taken λ = 1, κ̂2 = 6 and m̂2b = −0.5, −1.5, −1.75 for the black, blue and red lines respectively. The solid (dashed) line indicate the lightest (heaviest) mode. The quartic coupling λd varies along the horizontal axis according to eq. (B.5). The vertical line marks where λd = 0, having λd > 0 in the region to the right of it. while for β 2 /(λd λt )  1 these are κ̂2 β 2 m2φd ' hφd i2 , m2φt ' β hφd i2 . (B.7) 6 2λt With an abuse of notation, we have identified the lightest of the two modes as the dilaton, even though the eigenmodes corresponding to eq. (B.7) can have a sizeable mixing in the φd − φt basis. We show the full dependence on the parameters in figure 9, where we keep λt and β fixed and vary λd for various values of m̂2b (that is, of λt and β). One clearly sees several features: i) the solutions ‘start’ at ln(µc /hφd i) = 1/4 where the dilaton is massless, correspond- ing to the inflection point. This requires however λd < 0 that we already said is inapplicable. ii) the dilaton mass is suppressed by one power of β for λt λd ∼ O(1) (eq. (B.6)). In our model β can be small only near m̂2b = −2. In that case the suppression reads m2φd ∼ (m̂2b + 2)2 . Keep in mind, however, that m̂2b = −2 is not protected by any symmetry, so this is not representative of the full allowed parameter space. iii) the shapes of the lines resemble qualitatively those of figure 5. Nevertheless, the agreement between this and the 5D model is only expected for small m̂2 (φd ), since this measures how large is the tachyon VEV. A similar analysis can be done with more general choices of m̂2 (φd ) and the same qualitative behavior is observed quite generically as long as m̂2 (φd ) changes sign and has a moderate dependence on φd . Interestingly enough, it suffices to take that m̂2 (φd ) goes to a constant as φd → 0, in order to obtain a dilaton mass with a rising-decreasing shape as in figure 5 When the dilaton is lighter than the tachyon, for example for β  1, we can alterna- tively integrate out the tachyon from eq. (B.1) and obtain eq. (1.1) with λd β2 2 λeff (φd ) = − ln (φd /µc ) , (B.8) 4 4λt – 30 – that tells us that the explicit breaking of scale invariance is logarithmic, as expected from the dual theory due to the presence of the double-trace marginal operator Og . Eq. (1.1) with eq. (B.8) leads to eq. (B.5) and to the dilaton mass of eq. (B.6). As expected the dilaton mass is proportional to β2 βλeff (hφd i) = − ln(hφd i/µc ) ∼ β , (B.9) 2λt where in the last equality we have used eq. (B.5) with β  1. JHEP10(2019)202 B.1 Effective quartic coupling for the tachyon The quartic self-coupling for the 4D tachyon can be obtained readily by plugging into the 5D potential quartic term the normalized profile of the 5D tachyon field near the c → 0 condensation point and performing the integral over z. In the limit  → 0, zUV /zIR √ c with  ln (zUV /zIR ) finite, one obtains 9 + 2m̂2b   3 λt = + λ. (B.10) 8 2(10 + 6m̂2b + m̂4b )2 Even without any quartic self-coupling λ, the tachyon field experiences a stabilizing effect from its coupling to the metric. This is manifest in the background equation eq. (A.2), because the ‘friction’ term which depends on φ itself, see eq. (A.3). More explicitly, the metric can be integrated out by using eq. (A.3) to obtain a closed equation for φ v ! u u 1 κ̂2 L2 φ̇2 φ̈ = 4 t 2 + − V (φ) φ̇ + V 0 (φ) . 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