arXiv subjects

Luis E. Ibáñez

Publications and source records attributed to Luis E. Ibáñez.

At least 19 recordsLinked to original sources

arXiv2026

Some Properties and Uses of the Species Scale

The 'Species Scale' has proved to be an important concept when studying consistent effective actions in Quantum Gravity. This is a short summary of my contribution to the Corfu Summer Institute in September 2025, in which I covered two topics, both related in different ways to the fact that the Species Scale is moduli dependent. In the first, based on work done in collaboration with C. Aoufia and A. Castellano, we show how the one-loop Wilson coefficients $\mathcal{F}_n^{(d)}$ multiplyiing BPS protected ${\cal R}^{2n}$ operators obey Laplace-like eigenvalue differential equations of the form $\mathcal{D}^2_{\bf {\cal M}} \mathcal{F}^{(d)}_n = η_d\, \mathcal{F}^{(d)}_n$. This is true both for $n=2$ with 32 and 16 SUSY generators in 10,9,8 dimensions and theories with 8 SUSY generators in 6,5,4 dimensions $(n=1)$. We argue that this fact is at the root of some Swampland conjectures put forward in the past, like bounds on the dumping rates for the tower scales and the exponential behaviour in the Swampland Distance Conjecture. For the second topic, based on work done in collaboration with G.F. Casas, we discuss the one loop potential of the no-scale moduli in GKP-like Type IIB 4d orientifolds. To compute this potential we sum both over light and heavy (tower) modes using the Species Scale as a UV cut-off. We find a generic form $V_{1-loop}\sim g^2m_{3/2}^2M_p^2(g^{i{\bar i}}(\partial_iΛ)(\partial_{\bar i}Λ))/Λ^2$, with $Λ$ the Species Scale. This has minima at the $desert$ $points$ in moduli space and exponentially decreases at large moduli, with a dS hill in between. We argue that this potential may lead to the stabilisation of some or all Kahler moduli at the desert points in 4d Type IIB orientifolds of phenomenological interest.

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arXiv2025

Moduli Self-Fixing

In Quantum Gravity (QG), large moduli values lead to towers of exponentially light states, making the QG cut-off field-dependent. In 4D supersymmetric (SUSY) theories, this cut-off is set by the species scale $Λ(z_i, \bar{z}_i)$, where $z_i$ are complex moduli. We argue that in GKP-like 4D no-scale vacua, accounting for this field dependence generates one-loop, positive-definite potentials for the otherwise unfixed moduli, with local Minkowski minima at the desert points in moduli space with $z_i \sim \mathcal{O}(1)$. As these no-scale moduli grow, a dS plateau emerges and, for larger moduli, the potential runs away to zero, consistent with Swampland expectations. This may have important consequences for the moduli fixing problem. In particular non-perturbative superpotentials may not be necessary for fixing the Kähler moduli in a Type IIB setting. Although one loses control over non-perturbative corrections, we argue that modular invariance (more generally, dualities) of the species scale may give us information about the behaviour at small moduli in some simple cases. We illustrate this mechanism in a Type II 4D $\mathbb{Z}_2 \times \mathbb{Z}_2$ toroidal orientifold, where in some cases, modular invariance can give us control over the non-perturbative corrections. The vanishing cosmological constant reflects a delicate cancellation between IR and UV contributions at the minimum. The models may be completed by the addition of intersecting D6-branes, yielding a 3-generation MSSM (RR tadpole free) model with some (but not all) closed string moduli fixed in Minkowski, with the complex structure moduli fixed at the desert points.

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arXiv2024

On the Species Scale, Modular Invariance and the Gravitational EFT expansion

The concept of the species scale as the quantum gravity cut-off has been recently emphasised in the context of the Swampland program. Along these lines, we continue the quest for a precise understanding of its role within effective field theories of gravity as well as a global definition of the latter in case there is enough supersymmetry preserved. To do so, we exploit duality symmetries, the familiar asymptotic dependence imposed by the presence of infinite towers of light states and the known behaviour of higher-curvature corrections to the Einstein-Hilbert action in various String Theory setups. In those cases, we obtain a self-consistent result for the identification of the species scale as the quantum gravity cut-off, but also present some puzzles related to the suppression of certain higher-dimensional operators as well as minor ambiguities that may arise in the deep interior of moduli space.

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arXiv2024

Modular Invariant Starobinsky Inflation and the Species Scale

Potentials in cosmological inflation often involve scalars with trans-Planckian ranges. As a result, towers of states become massless and their presence pushes the fundamental scale not to coincide with $M_{\rm P}$ but rather with the $species\, scale$, $Λ$. This scale transforms as an automorphic form of the theory's duality symmetries. We propose that the inflaton potential should be 1) an automorphic invariant form, non-singular over all moduli space, 2) depending only on $Λ$ and its field derivatives, and 3) approaching constant values in the large moduli to ensure a long period of inflation. These conditions lead to the proposal $V \sim λ(ϕ, ϕ^*)$, with $λ= G^{i\bar{j}} (\partial_i Λ)(\partial_{\bar{j}} Λ) / Λ^2$, determining the 'species scale convex hull'. For a single elliptic complex modulus with $SL(2, Z)$ symmetry, this results in an inflaton potential $V \simeq (\text{Im} τ)^2 |\tilde{G}_2|^2 / N^2$, with $N \simeq -\log(\text{Im} τ|η(τ)|^4)$, where $η$ is the Dedekind function and $\tilde{G}_2$ the Eisenstein modular form of weight 2. Surprisingly, this potential at large modulus resembles that of the Starobinsky model. We compute inflation parameters yielding results similar to Starobinsky's, but extended to modular invariant expressions. Interestingly, the number of e-folds is proportional to the number of species in the tower, $N_e \simeq N$, and $ε\simeq Λ^4$ at large moduli, suggesting that the tower of states plays an important role in the inflation process.

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arXiv2024

Yukawa Couplings at Infinite Distance and Swampland Towers in Chiral Theories

We study limits of vanishing Yukawa couplings of 4d chiral matter fields in Quantum Gravity, using as a laboratory type IIA orientifolds with D6-branes. In these theories chiral fermions arise at brane intersections, where an infinite tower of charged particles dubbed gonions are localised. We show that in the limit $Y\rightarrow 0$ some of these towers become asymptotically massless, while at the same time the kinetic term of some chiral fields becomes singular and at least two extra dimensions decompactify. For limits parametrised by a large complex structure saxion $u$, Yukawa couplings have a behaviour of the form $Y \, \sim \, {1}/{u^r}$, with $r$ some positive rational number. Moreover, in this limit some of the gauge couplings associated to the Yukawa vanish. The lightest gonion scales are of order $m_{\rm gon} \sim g^s M_{\rm P}$ with $s>1$, verifying the magnetic WGC with room to spare and with no need of its tower/sublattice versions. We also show how this behaviour can be understood in the context of the emergence of kinetic terms in Quantum Gravity. All these results may be very relevant for phenomenology, given the fact that some of the Yukawa couplings in the Standard Model are very small.

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arXiv2024

On small Dirac Neutrino Masses in String Theory

We study how tiny Dirac neutrino masses consistent with experimental constraints can arise in string theory SM-like vacua. We use as a laboratory 4d ${\cal N}=1$ type IIA Calabi--Yau orientifold compactifications, and in particular recent results on Yukawa couplings at infinite field-space distance. In this regime we find Dirac neutrino masses of the form $m_ν\simeq g_ν\langle H\rangle$, with $g_ν$ the gauge coupling of the massive $U(1)$ under which the right-handed neutrinos $ν_R$ are charged, and which should be in the range $g_ν\simeq 10^{-14}-10^{-12}$ to reproduce neutrino data. The neutrino mass suppression occurs because the right-handed neutrino kinetic term behaves as $K_{νν} \simeq 1 /g_ν^2 $. At the same time a tower of $ν_R$-like states appears with characteristic scale $m_0\simeq g_ν^2M_{\rm P}\simeq 0.1-500$ eV, in agreement with Swampland expectations. Two large hidden dimensions only felt by the $ν_R$ sector arise at the same scale, while the string scale is around $M_s\simeq g_νM_{\rm P}\simeq 10-700$ TeV. Some phenomenological implications and model building challenges are described. As a byproduct, independently of the neutrino issue, we argue that a single large dimension in the context of SM-like type IIA Calabi--Yau orientifolds leads to too small Yukawa couplings for quarks and charged leptons.

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arXiv2024

Entropy Bounds and the Species Scale Distance Conjecture

The Swampland Distance Conjecture (SDC) states that, as we move towards an infinite distance point in moduli space, a tower of states becomes exponentially light with the geodesic distance in any consistent theory of Quantum Gravity. Although this fact has been tested in large sets of examples, it is fair to say that a bottom-up justification that explains both the geodesic requirement and the exponential behavior has been missing so far. In the present paper we address this issue by making use of the Covariant Entropy Bound as applied to the EFT. When applied to backgrounds of the Dynamical Cobordism type in theories with a moduli space, we are able to recover these main features of the SDC. Moreover, this naturally leads to universal lower and upper bounds on the 'decay rate' parameter $λ_{\text{sp}}$ of the species scale, that we propose as a convex hull condition under the name of Species Scale Distance Conjecture (SSDC). This is in contrast to already proposed universal bounds, that apply to the SDC parameter of the lightest tower. We also extend the analysis to the case in which asymptotically exponential potentials are present, finding a nice interplay with the asymptotic de Sitter conjecture. To test the SSDC, we study the convex hull that encodes the (asymptotic) moduli dependence of the species scale. In this way, we show that the SSDC is the strongest bound on the species scale exponential rate which is preserved under dimensional reduction and we verify it in M-theory toroidal compactifications.

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arXiv2023

Towers and Hierarchies in the Standard Model from Emergence in Quantum Gravity

Based on Quantum Gravity arguments, it has been suggested that all kinetic terms of light particles below the UV cut-off could arise in the IR via quantum (loop) corrections. These loop corrections involve infinite towers of states becoming light (e.g. Kaluza-Klein or string towers). We study implications of this Emergence Proposal for fundamental scales in the Standard Model (SM). In this scheme all Yukawa couplings are of order one in the UV and small Yukawas for lighter generations appear via large anomalous dimensions induced by the towers of states. Thus, the observed hierarchies of quark and lepton masses are a reflection of the structure of towers of states that lie below the Quantum Gravity scale, $Λ_{\text{QG}}$. Small Dirac neutrino masses consistent with experimental observation appear due to the existence of a tower of SM singlet states of mass $m_{0}\simeq Y_{ν_3}M_p\simeq 7\times 10^5$ GeV, opening up a new extra dimension, while the UV cut-off occurs at $Λ_{\text{QG}}\lesssim 10^{14}$ GeV. Additional constraints relating the Electro-Weak (EW) and cosmological constant (c.c.) scales (denoted $M_{\text{EW}}$ and $V_0$) appear if the Swampland condition $m_{ν_1}\lesssim V_0^{1/4}$ is imposed (with $ν_1$ denoting the lightest neutrino), which itself arises upon applying the AdS non-SUSY Conjecture or the AdS/dS Distance Conjecture to the 3d vacua from circle compactifications of the SM. In particular, the EW scale and that of the extra dimension fulfill ${m_0 \, M_{\text{EW}}\lesssim 10^{2}\, V_0^{1/4}M_p}$, thus relating the EW hierarchy problem to that of the c.c. Hence, all fundamental scales may be written as powers of the c.c., i.e. $m_{\bullet}\sim V_0^δM_p^{1-4δ}$. The scale of SUSY breaking is $m_{3/2}\lesssim 7\times 10^5$ GeV, which favours a Mini-Split scenario that could be possibly tested at LHC and/or FCC.

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arXiv2023

The Emergence Proposal in Quantum Gravity and the Species Scale

In the Emergence Proposal in QG it is conjectured that all light-particle kinetic terms are absent in the fundamental UV theory and are generated by quantum corrections in the IR. It has been argued that this may provide for some microscopic understanding of the WGC and SDC. In the present paper we take the first steps towards a systematic study of Emergence in the context of string theory. We emphasize the crucial role of the species scale in any EFT coupled to gravity, and discuss its computation in string theory and general systems with light towers of states. We then introduce the notion of Emergence and show how kinetic terms for moduli, gauge bosons and fermions may be generated. One-loop computations play an important role in Emergence, so we present detailed calculations in $d$ spacetime dimensions for the wave-function renormalization of scalars, vectors and fermions. We extend and check the Emergence Proposal in a number string vacua, including 4d $\mathcal{N}=2$ theories arising from type IIA on a CY$_3$, where the towers at strong coupling are comprised by D0 and (wrapped) D2-branes, and also elaborate on how instanton corrections would fit within the emergence picture. Higher dimensional examples are also discussed, including 6d and 7d models arising from F-/M-theory on an elliptic CY$_3$ or a $K3$ surface. We also consider 10d string theories and study in some detail the emergence mechanism in type IIA. We show as well how the flux potential in 4d may be obtained from the emergence prescription, by analyzing the corresponding decompactification limits to M-theory. We find that the required kinetic terms for the dual 3-form fields can arise upon integrating out towers of massive gravitini (and bosonic superpartners). Our analysis renders support to the Emergence Proposal, and to the idea that infinite distance singularities may arise in QG as an intrinsic IR phenomenon.

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arXiv2022

IR/UV Mixing, Towers of Species and Swampland Conjectures

By applying the Covariant Entropy Bound (CEB) to an EFT in a box of size $1/Λ_{\text{IR}}$ one obtains that the UV and IR cut-offs of the EFT are necessarily correlated. We argue that in a theory of Quantum Gravity (QG) one should identify the UV cutoff with the `species scale', and give a general algorithm to calculate it in the case of multiple towers becoming light. One then obtains an upper bound on the characteristic mass scale of the tower in terms of the IR cut-off, given by $M_{\text{tower}}\lesssim (Λ_{\text{IR}})^{2α_D}$ in Planck units, with $α_D=(D-2+p)/2p(D-1)$, where $p$ depends on the density of states. Identifying the IR cut-off with a (non-vanishing) curvature in AdS one reproduces the statement of the AdS Distance Conjecture (ADC), also giving an explicit lower bound for the $α$ exponent. In particular, we find that the CEB implies $α\geq1/2$ in any dimension if there is a single KK tower, both in AdS and dS vacua. However values $α<1/2$ are allowed if the particle tower is multiple or has a string component. We also consider the CKN constraint coming from avoiding gravitational collapse which further requires in general $α\geq 1/D$ for the lightest tower. We analyse the case of the DGKT-CFI class of Type IIA orientifold models and show it has both particle and string towers below the species scale, so that a careful analysis of how the ADC is defined is needed. We find that this class of models obey but do not saturate the CEB. The UV/IR constraints found apply to both AdS and dS vacua. We comment on possible applications of these ideas to the dS Swampland conjecture as well as to the observed dS phase of the universe.

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arXiv2021

AdS Swampland Conjectures and Light Fermions

We consider constraints on $D$-dimensional theories in $M_D$, $dS_D$ and $AdS_D$ backgrounds in the light of AdS swampland conjectures as applied to their compactification in a circle. In particular we consider the non-SUSY AdS instability conjecture and the AdS distance conjecture. For $M_D$ and $dS_D$ vacua the results may be summarized by a light fermion conjecture which states that in theories with $Λ_D\geq 0$ and a positive first non-vanishing supertrace $(-1)^{k+1}\text{Str} M^{2k}>0$, a surplus of light fermions with mass $m_f\lesssim Λ_D^{1/D}$ must be present. On the contrary, the cases of $M_D$ and $AdS_D$ can be made consistent with the mild but not the strong version of the AdS Distance conjecture, since the KK tower in the lower $d$-dim theory will scale as $m\lesssim Λ_d^α$ with $α=1/d$. The above constraints also suggest that the Standard Model of particle physics would be inconsistent in Minkowski space but consistent in dS if the lightest neutrino is Dirac and lighter than the cosmological constant scale.

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arXiv2021

A Gravitino Distance Conjecture

We conjecture that in a consistent supergravity theory with non-vanishing gravitino mass, the limit $m_{3/2}\rightarrow 0$ is at infinite distance. In particular one can write $M_{\mathrm{tower}} \sim m_{3/2}^δ$ so that as the gravitino mass goes to zero, a tower of KK states as well as emergent strings becomes tensionless. This conjecture may be motivated from the Weak Gravity Conjecture as applied to strings and membranes and implies in turn the AdS Distance Conjecture. We test this proposal in classical 4d type IIA orientifold vacua in which one obtains a range of values $\tfrac13 \le δ\le 1$. The parameter $δ$ is related to the scale decoupling exponent in AdS vacua and to the $α$ exponent in the Swampland Distance Conjecture for the type IIA complex structure. We present a general analysis of the gravitino mass in the limits of moduli space in terms of limiting Mixed Hodge Structures and study in some detail the case of two-moduli F-theory settings. Moreover, we obtain general lower bounds $δ\, \geq \, \frac{1}{3}, \, \frac{1}{4}$ for Calabi--Yau threefolds and fourfolds, respectively. The conjecture has important phenomenological implications. In particular we argue that low-energy supersymmetry of order 1 TeV is only obtained if there is a tower of KK states at an intermediate scale, of order $10^8$ GeV. One also has an upper bound for the Hubble constant upon inflation $H \lesssim m_{3/2}^δM^{(1-δ)}_{\text{P}}$.

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arXiv2020

Pair Production and Gravity as the Weakest Force

The Weak Gravity Conjecture (WGC) is usually formulated in terms of the stability of extremal black-holes or in terms of long distance Coulomb/Newton potentials. However one can think of other physical processes to compare the relative strength of gravity versus other forces. We argue for an alternative formulation in terms of particle pair production at threshold or, equivalently, pair annihilation at rest. Imposing that the production rate by any force mediator (photon or scalar) of pairs of charged particles be larger or equal to graviton production, we recover known conditions for the $U(1)$ WGC and its extensions. Unlike other formulations though, threshold pair production is sensitive to short range couplings present in scalar interactions and gives rise to a Scalar WGC. Application to moduli scalars gives rise to specific conditions on the trilinear and quartic couplings which involve first and second derivatives of the WGC particle mass with respect to the moduli. Some extremal solutions to these equations correspond to massive states behaving like BPS, KK and winding states which feature duality invariance and are in agreement with the Swampland distance conjecture. Conditions for $N=2$ BPS states saturate our bounds and we discuss specific examples of BPS states which become massless at large Kahler moduli in Type IIA N=2, D=4 CY and orbifold compactifications. We study possible implications for potentials depending on moduli only through WGC massive states. For some simple classes of potentials one recovers constraints somewhat similar but not equivalent to a Swampland dS conjecture.

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arXiv2020

On scale separation in type II AdS flux vacua

We study the separation of AdS and Kaluza-Klein (KK) scales in type II 4d AdS orientifold vacua. We first address this problem in toroidal/orbifold type IIA vacua with metric fluxes, corresponding to compactifications in twisted tori, both from the 4d and 10d points of view. We show how the naive application of the effective 4d theory leads to results which violate the AdS distance conjecture, in a class of $\mathcal{N}=1$ supersymmetric models which have a 10d lifting to a compactification on $S^3\times S^3$. We show how using KK scales properly modified by the compact metric leads to no separation of scales with $M^2_{\text{KK}} = \mathfrak{c} |Λ|$, with $\mathfrak{c}$ a numerical constant independent of fluxes. This applies with no need to keep non-leading fluxes fixed. We also consider a class of IIB models with non-geometric fluxes in which the effective field theory analysisseems to lead to a naive separation of scales and a violation of the AdS distance conjecture. It has a T-dual which again may be understood as a 10d type IIA theory compactified on $S^3\times S^3$. In this geometric dual one again observes that the strong AdS distance conjecture is obeyed with $M^2_{\text{KK}} = \mathfrak{c}' |Λ|$, if one takes into account the curvature in the internal space. These findings seem to suggest that all toroidal/orbifold models with fluxes in this class obey $M^2_{\text{KK}} = \mathfrak{c} |Λ|$ with $\mathfrak{c}$ a flux-independent numerical constant.

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arXiv2019

The Swampland Distance Conjecture and Towers of Tensionless Branes

The Swampland Distance Conjecture states that at infinite distance in the scalar moduli space an infinite tower of particles become exponentially massless. We study this issue in the context of 4d type IIA and type IIB Calabi-Yau compactifications. We find that for large moduli not only towers of particles but also domain walls and strings become tensionless. We study in detail the case of type IIA and IIB ${\cal N}=1$ CY orientifolds and show how for infinite Kähler and/or complex structure moduli towers of domain walls and strings become tensionless, depending on the particular direction in moduli space. For the type IIA case we construct the monodromy orbits of domain walls in detail. We study the structure of mass scales in these limits and find that these towers may occur at the same scale as the fundamental string scale or the KK scale making sometimes difficult an effective field theory description. The structure of IIA and IIB towers are consistent with mirror symmetry, as long as towers of exotic domain walls associated to non-geometric fluxes also appear. We briefly discuss the issue of emergence within this context and the possible implications for 4d vacua.

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arXiv2019

A Strong Scalar Weak Gravity Conjecture and Some Implications

We propose a new version of the scalar Weak Gravity Conjecture (WGC) which would apply to any scalar field coupled to quantum gravity. For a single scalar it is given by the differential constraint $V''\leq (2V'''^2/V''-V'''')M_{\text{p}}^2$. It corresponds to the statement that self-interactions of a scalar must be stronger than gravity for any value of the scalar field. We find that the solutions which saturate the bound correspond to towers of extremal states with mass $m^2(ϕ)=m_0^2/((R/m)^2+1/(nR)^2)$, with $R^2=e^ϕ$, consistent with the emergence of an extra dimension at large or small $R$ and the existence of extended objects (strings). These states act as WGC states for the scalar $ϕ$. It is also consistent with the distance swampland conjecture with a built-in duality symmetry. From this constraint one can derive several swampland conjectures. In particular one finds that an axion potential is only consistent if $f\leq M_{\text{p}}$. The conjecture has far reaching consequences and applies to several interesting physical systems: i) Among chaotic inflation potentials only those asymptotically linear may survive. ii) If applied to the radion of the circle compactification of the Standard Model to 3D, the constraint implies that the 4D cosmological constant scale must be larger than the mass of the lightest neutrino, which must be Dirac and in normal hierarchy. iii) It also constraints simplest moduli fixing string models. The simplest KKLT model is compatible with the constraints but the latter may be relevant for some choices of parameters.

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arXiv2019

Modular Symmetries and the Swampland Conjectures

Recent string theory tests of swampland ideas like the distance or the dS conjectures have been performed at weak coupling. Testing these ideas beyond the weak coupling regime remains challenging. We propose to exploit the modular symmetries of the moduli effective action to check swampland constraints beyond perturbation theory. As an example we study the case of heterotic 4d $\mathcal{N}=1$ compactifications, whose non-perturbative effective action is known to be invariant under modular symmetries acting on the Kähler and complex structure moduli, in particular $SL(2,Z)$ T-dualities (or subgroups thereof) for 4d heterotic or orbifold compactifications. Remarkably, in models with non-perturbative superpotentials, the corresponding duality invariant potentials diverge at points at infinite distance in moduli space. The divergence relates to towers of states becoming light, in agreement with the distance conjecture. We discuss specific examples of this behavior based on gaugino condensation in heterotic orbifolds. We show that these examples are dual to compactifications of type I' or Horava-Witten theory, in which the $SL(2,Z)$ acts on the complex structure of an underlying 2-torus, and the tower of light states correspond to D0-branes or M-theory KK modes. The non-perturbative examples explored point to potentials not leading to weak coupling at infinite distance, but rather diverging in the asymptotic corners of moduli space, dynamically forbidding the access to points with global symmetries. We perform a study of general modular invariant potentials and find that there are dS maxima and saddle points but no dS minima, and that all examples explored obey the refined dS conjecture.

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arXiv2018

The Fundamental Need for a SM Higgs and the Weak Gravity Conjecture

Compactifying the SM down to 3D or 2D one may obtain AdS vacua depending on the neutrino mass spectrum. It has been recently shown that, by insisting in the absence of these vacua, as suggested by {\it Weak Gravity Conjecture} (WGC) arguments, intriguing constraints on the value of the lightest neutrino mass and the 4D cosmological constant are obtained. For fixed Yukawa coupling one also obtains an upper bound on the EW scale $\left\langle H\right\rangle\lesssim {Λ_4^{1/4}} /{Y_{ν_{i}}}$,where $Λ_4$ is the 4D cosmological constant and $Y_{ν_{i}}$ the Yukawa coupling of the lightest (Dirac) neutrino. This bound may lead to a reassessment of the gauge hierarchy problem. In this letter, following the same line of arguments, we point out that the SM without a Higgs field would give rise to new AdS lower dimensional vacua. Absence of latter would require the very existence of the SM Higgs. Furthermore one can derive a lower bound on the Higgs vev $\left\langle H\right\rangle\gtrsim Λ_{\text{QCD}}$ which is required by the absence of AdS vacua in lower dimensions. The lowest number of quark/lepton generations in which this need for a Higgs applies is three, giving a justification for family replication. We also reassess the connection between the EW scale, neutrino masses and the c.c. in this approach. The EW fine-tuning is here related to the proximity between the c.c. scale $Λ_4^{1/4}$ and the lightest neutrino mass $m_{ν_i}$ by the expression $ \frac {ΔH}{H} \lesssim \frac {(aΛ_4^ {1/4} -m_{ν_i})} {m_{ν_i}}. $ We emphasize that all the above results rely on the assumption of the stability of the AdS SM vacua found.

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