Constraining Neutrino Masses, the Cosmological Constant and BSM Physics from the Weak Gravity Conjecture

It is known that there are AdS vacua obtained from compactifying the SM to 2 or 3 dimensions. The existence of such vacua depends on the value of neutrino masses through the Casimir effect. Using the Weak Gravity Conjecture, it has been recently argued by Ooguri and Vafa that such vacua are incompatible with the SM embedding into a consistent theory of quantum gravity. We study the limits obtained for both the cosmological constant $\Lambda_4$ and neutrino masses from the absence of such dangerous 3D and 2D SM AdS vacua. One interesting implication is that $\Lambda_4$ is bounded to be larger than a scale of order $m_\nu^4$, as observed experimentally. Interestingly, this is the first argument implying a non-vanishing $\Lambda_4$ only on the basis of particle physics, with no cosmological input. Conversely, the observed $\Lambda_4$ implies strong constraints on neutrino masses in the SM and also for some BSM extensions including extra Weyl or Dirac spinors, gravitinos and axions. The upper bounds obtained for neutrino masses imply (for fixed neutrino Yukawa and $\Lambda_4$) the existence of upper bounds on the EW scale. In the case of massive Majorana neutrinos with a see-saw mechanism associated to a large scale $M\simeq 10^{10-14}$ GeV and $Y_{\nu_1}\simeq 10^{-3}$, one obtains that the EW scale cannot exceed $M_{EW}\lesssim 10^2-10^4$ GeV. From this point of view, the delicate fine-tuning required to get a small EW scale would be a mirage, since parameters yielding higher EW scales would be in the swampland and would not count as possible consistent theories. This would bring a new perspective into the issue of the EW hierarchy.


Introduction
The deep infrared region of the Standard Model (SM), the region below the electron mass m e threshold, is quite simple. It only includes a few bosonic degrees of freedom (2 from the photon and 2 from the graviton) and a few fermionic degrees of freedom ( It has always been intriguing the proximity of the c.c. scale to that of neutrino masses since both scales seem to have a very different origin. The c.c. comes from the vacuum energy of the SM and its smallness is a major puzzle in the theory. One possible explanation of the smallness of the c.c. is the existence of a landscape of vacua, as in string theory [1,2], with this small value required for the development of galaxy formation, as first suggested by Weinberg [3]. On the other hand the smallness of neutrino masses arises naturally from a see-saw mechanism, if neutrinos are Majorana, whereas it is less natural to attain so small neutrino masses in the Dirac case. It would be interesting to find links between the values of the Λ 1/4 0 and m ν scales which are quite close and around 7 orders of magnitude smaller than the next higher mass scale given by the electron mass m e . In ref. [4] Arkani-Hamed et al explored this deep infrared SM region by making the interesting exercise of exploring the possible vacua that could be obtained by compactifying the SM action to lower dimensions. They found that there is a richness of vacua, a real landscape of vacua, both with AdS and dS geometries in 3D and 2D (see also [5,6] for low < 3D SM compactifications). The potential for the moduli of the compactification is induced by the Casimir effect of the lightest particles of the SM.
The existence or not of these lower dimensional vacua turns out to depend sensitively on the value of neutrino masses. For example, they found that if all neutrinos are Majorana and we compactify down to 3 or 2 dimensions AdS SM vacua do appear for any values of neutrino masses consistent with experiment. Interestingly, these vacua are almost identical to the 4D SM at distances larger than 20 microns or so. Still, these 3D,2D vacua look like a curiosity with no real physical relevance to our world.
In an apparently very unrelated development, there has been in the last few years a renewed interest in the Weak Gravity Conjecture (WGC) [7]. In simplified terms the WGC states that in any consistent theory of quantum gravity, the gravitational interaction must be always weaker than any other interaction. This statement is motivated by blackhole arguments and has been shown to be a powerful criterium to determine whether an effective field theory has a consistent UV completion (see [8][9][10] for some recent applications of the WGC). The main support for the WGC comes from the fact that no contradiction has been found with any string theory example. Recently a sharpened version of the WGC has been put forward by Ooguri and Vafa [11] with a quite restrictive corollary (see also [12] for related work). It states that no stable non-SUSY AdS vacua can be consistent with quantum gravity. They also note in passing that if the AdS SM lower dimensional vacua of [4] exist and are stable, then the 4D SM itself could not be completed in the UV. In particular a minimal setting with Majorana neutrinos would be ruled out.
This is a very interesting remark. The weakest point in the argumentation is that, even if we take for granted this sharpened WGC, it applies only if the said 3D,2D vacua are actually stable. If they are unstable, no inconsistency with quantum gravity would appear, which would result in no constraints. Some potential instabilities (like decay into Witten's bubble of nothing) are not present in these SM vacua, due to the periodic boundary conditions of the fermions. However one may argue that other potential instabilities may appear e.g. in the context of a 4D landscape of vacua in string theory, in which tunneling in 4D would have parallel transitions in lower dimensions, rendering the lower dimensional vacua unstable, and hence leading to no constraint on neutrino masses or any other physical parameter.
In this paper we reanalyze the issue of the possible constrains on neutrino masses but also on the value of the c.c. from the assumption that no lower dimensional AdS vacua of the SM should exist. We also do this analysis if additional light BSM particles (axions, Weyl fermions, Dirac fermions) are present well bellow the electron mass threshold. We are aware that this assumption may be unjustified, since the stability of the AdS lower dimensional SM is far from clear, as we will discuss in the text. Still our knowledge of the 4D landscape of vacua is very poor and their stability is not excluded. Furthermore, the fact that this assumption leads to intriguing connections between the c.c., the neutrino masses and possible additional very light degrees of freedom make this study worthwhile. In fact we find quite amazing that a simple very abstract condition like the absence of lower dimensional SM AdS vacua leads to interesting and potentially testable conditions on the infrared degrees of freedom of the SM. One would have expected that such abstract condition would had lead to totally wild predictions, and we rather find conditions which are close to be fulfilled by the SM or some simple extension. In our analysis we confirm that a simplest scheme with 3 Majorana neutrinos would be ruled out within this scheme. However the addition e.g. of a single very light Weyl fermion to the SM makes the Majorana possibility viable. Dirac neutrinos are viable for the lightest neutrino light enough. So e.g. a potential measurement of (natural hierarchy) Majorana masses at ν-less double βdecay experiments would then imply some additional BSM physics like the existence of additional very light sterile neutrinos.
We also analyze in detail the role of the 4D c.c. on the constraints and find that the 4D c.c. has a lower bound depending on the value of neutrino masses. As the c.c.
grows above the neutrino mass scale, the easier is to avoid that AdS vacua develop.
This is important because it is the first argument for a non-vanishing Λ 4 based only on particle physics and not on cosmology. The structure of this paper is as follows. In the next section we briefly review the Weak Gravity Conjecture in connection with AdS vacua discussed in [11]. We also critically discuss the issue of the instability of the in order to allow for (sub)extremal black holes to decay and avoid the usual trouble with remnants. In the last years there has been a lot of progress generalising the conjecture for multiple gauge fields and applying it to inflation [8][9][10]. However, a proof has not been found yet, and the strongest evidence for the conjecture in fact comes from the lack of a counter-example in string theory up to now. Ooguri and Vafa proposed in [11] a sharpened version of this conjecture, claiming that the equality can only be satisfied if the charged object is BPS and the theory is supersymmetric. This has dramatic consequences for the AdS/CFT duality as we review in the following. It implies that any non-supersymmetric AdS vacuum supported by fluxes must contain a membrane charged under the flux whose tension is smaller than its charge. If this is the case, the possibility of nucleating such a membrane corresponds to an instability of the vacuum, as shown by Maldacena et al. in [13] (see also [14,15] sions. If these vacua are stable, they would be inconsistent with the above conjecture.
As Ooguri and Vafa commented in [11], this would rule out the possibility of Majorana neutrinos in the SM. Before turning to a more thoughtful analysis of these constraints, let us comment, though, on the issue of stability. if the bubble radius in four dimensions is smaller than the dS 4 length but still bigger than the AdS 3 length, i.e. l 3 < R b < l 4 .

Instabilities in the landscape
Let us compute how big is the stability window for the case at hand. The dS 4 length scale in our universe is given by where we have used that the cosmological constant is V 0 = 2.6 · 10 −47 GeV 4 . Upon compactifying on a circle of radius R, the AdS 3 length of the resulting three dimensional space reads However, it can be made quite large for the case of heavier neutrinos, still consistent with the Planck cosmological bound. In overall, the AdS 3 length can vary between zero and two orders of magnitude. Therefore, instabilities in four dimensions described by a growing bubble whose size is of order 0.01 l 4 R b l 4 will not yield instabilities in three dimensions. The question now is, in which cases will the membranes mediating vacuum decay have such a critical radius?
Let us first assume that the instability in four dimensions can be described by a Coleman-De Luccia (CDL) instanton within the thin-wall approximation [16]. The size of the bubble is given by where γ = (κT ) 2

4
− ∆Λ i and ∆Λ = Λ i − Λ f . We also use the standard notation for the cosmological constant Λ i = κV i /3 with κ = 1/M 2 p . There are two interesting limits depending on whether gravitational effects are important (T ∆Λ) yielding Since Λ i in our universe is very small, this relation has to be satisfied with a high accuracy. More concretely, if ≡ T 2 − 4M 4 p ∆Λ one needs κ /(4T ) Λ i . As explained above, this is the largest radius the bubble can have in deSitter, and gives rise to a very suppressed tunneling rate at the edge of stability. Intuitively, it corresponds to the case in which the SM is separated from other vacua in the landscape by huge potential barriers. Furthermore, in a supersymmetric theory it corresponds to the BPS bound (static domain walls are given indeed by BPS membranes). Since we are in a non-supersymmetric configuration, the membrane action might receive corrections that bring it away from the above bound. If those corrections go in the direction of decreasing the tension T < M 2 p Q (in a way consistent with the WGC above) and supersymmetry is only slightly broken, one might expect that the condition (2.8) is still approximately satisfied. In such a case, these membranes would fit in the window l 3 < R b < l 4 and the 3d vacuum would be stable. However, any quantitative analysis is model dependent and out of reach at present.
On the other hand, such a radius is characteristic of a Hawking-Moss (HM) process (see e.g. [17][18][19][20][21] and references therein). A HM instanton describes the quantum transition of a field starting at the minimum and emerging at the top of the barrier due to quantum fluctuations in a sort of Brownian motion [19][20][21]. After emerging, the field will roll down the potential until the next minimum. This process allows to connect minima for which a CDL instanton does not exist, and has been argued to be essential to populate the landscape [21], since up-tunneling from AdS cannot proceed through usual CDL instantons. The decay rate of this stochastic diffusion process is equivalent to that of an homogeneous tunneling of the whole universe in which the bubble radius is R b ∼ l 4 . A HM process will be dominant with respect to CDL whenever the thickness of the barrier is bigger than the height. Therefore, if the SM is separated from other vacua in the landscape by thick barriers, the corresponding 3d vacua might be stable.
Yet another possibility would be that the 4d vacuum is stable, but an instability appears upon compactification. The only known example of this type on a circle compactification is the Witten's bubble [22], which describes the decay of spacetime to nothing. However, this bubble is only topologically consistent with antiperiodic boundary conditions for the fermions around S 1 , while the AdS 3 vacuum exists only for periodic boundary conditions. Therefore, the bubble of nothing is not allowed in our case.
To summarize, it seems that the AdS 3 vacua will be stable unless the parent dS 4 is unstable and the corresponding bubble size is much smaller than the dS 4 length, so it does not lie in the range l 3 < R b < l 4 . Unfortunately, without a better understanding of the string landscape, we cannot argue one way or another. From now on, we will explore the consequences of assuming that the derived minima are stable. According to the Ooguri-Vafa conjecture, a stable non-supersymmetric AdS vacuum is not consistent with quantum gravity, which leads to interesting constraints on the SM matter spectrum to avoid the appearance of AdS minima upon compactification. We find interesting that the constraints derived in this way are close to the experimental bounds on neutrino masses for the observed value of the c.c.

AdS Casimir SM vacua in 3D
The conjecture of Ooguri and Vafa implies that no stable non-SUSY AdS SM vacua should exist. Assuming background independence, this statement should apply to any lower dimensional compactification of the SM. The simplest case is the 3D in which the SM is compactified on a circle, which we will discuss in this section. The compactifications down to 2D are richer, in the sense that there are more options, the simplest one being the compactification on a 2-torus, which we will study in the next section. Furthermore one can switch on electromagnetic fluxes through the torus, giving rise to a rich spectrum of vacua. More generally one can consider compactifications on general Riemann surfaces. Those have been argued in [4,6] not to lead to new vacua.
The same has also been argued to be the case of 1D vacua [5]) (quantum mechanics).
For these reasons we will concentrate in this paper on 3D vacua and 2D toroidal vacua with no fluxes, which are the only vacua in which the Casimir contribution plays an important role and can lead to constraints on the spectrum of neutrino masses and other BSM very light additional particles.

The radion potential in 3D
In this section we review the origin and numerics of the 3D SM vacua first discussed in [4]. The 3D action obtained upon compactification of the SM on a circle of radius R has the form Here M p is the 4D reduced Planck mass, M p = (8πG N ) −1/2 and Λ 4 is the 4D cosmological constant. The scale r is later to be fixed equal to the vev of the radion R. It also displays the action of the graviphoton with field strength W µν and the radion field R.
The action shows a runaway potential for the radion coming from the 4D cosmological However the 4D c.c. is so tiny that the quantum contribution of the lightest SM modes to the vacuum energy may become important for the computation of the radion potential. The 1-loop corrections to the effective potential of SM particles can be derived from the Casimir energy coming from loops wrapping the circle. For massive particles such contributions are exponentially supressed like ∝ e −2πmR for R 1/m.
This means that only particles with mass lighter than 1/R contribute significantly. In the case of massless particles the Casimir contribution to the potential becomes very simple. One obtains that is written in the Weyl-rescaled metric as, (3.4) The minus sign stands for bosons and the plus sign for fermions with periodic boundary conditions (minus for antiperiodic). The integer n 0 is the number of degrees of freedom of the particle (two for massless vector bosons, two for Majorana fermions, four for Dirac fermions, etc).
The only massless degrees of freedom in the SM+gravitation are 4 = 2 + 2 from the photon and the graviton. If we only take into account these contributions the effective potential reads, where the number four comes from the sum of the degrees of freedom of the massless particles. In Fig. 1 the contributions from the massless states and the cosmological constant are depicted. The contribution of the cosmological constant is shown as a black line. If we include the massless states, the graviton and the photon, we see that the effective potential, red line, drops for small R. In this case there is no minimum.
It is clear that the negative sign of the bosonic massless states pushes the effective potential to negative values for small radius due to the sixth power of the radion field, R −6 compared with the squared of the cosmological constant, R 2 , that is important for larger values of R. However a maximum appears due to the different behaviour of the two contributions. This maximum occurs at R max [4], Using the value of the cosmological constant, Λ 4 = 2.6 · 10 −47 GeV 4 [23], and the associated mass scale here will be, Interestingly, this scale is close to the neutrino mass scale. As we decrease the value of R, the next threshold in the SM is that of neutrino masses. With periodic boundary conditions for neutrinos, schematically the potential is modified as with R i = 1/m ν i and Θ a step function. As R decreases the different neutrino thresholds open and eventually overwhelm the bosonic contribution, giving rise to possible minima, as long as R i < R max . Minima turns out to develop at R 0 1/m ν where here m ν refers to the first threshold for which the number of fermionic degrees of freedom becomes larger than 4.
In practice a correct computation of the minima depends sensitively on the values of the neutrino masses and the cosmological constant, and a full computation of the Casimir contributions, including mass effects is required. In a general case for a particle of mass, m the Casimir energy density is given by [4] where K i (x) is the Bessel function. We will use this formula in the computation of the minima below. It is however interesting to expand this formula for small (mR), Summing the contributions of the cosmological constant and the particles the effective potential reads, where the sum goes over all the particles in the spectrum, n i is the number of degrees of freedom of the i-th particle and s i = 0(1) periodic fermions or bosons respectively.
One obtains a general formula for the potential in terms of the Weyl-rescaled metric for small masses Setting the scale r such that 2πr = 1 GeV −1 the effective potential is written, Note that this formula is not a good approximation to study the minima generated by neutrinos because, as we said, the minima are obtained at R 0 1/m ν and hence (Rm) is not in general small. However in the case of the lightest neutrino (or some additional very light BSM state) (Rm) may be small enough so that the dependence on these masses is adequately described by this expression. We will also use it as an inspiration to fit the curve which parametrises the lowest value of the cosmological constant required to get positive vacuum energy in section 3.5.

Limits on neutrino masses
As we have discussed in the previous chapter, we want to impose that no stable AdS We also do not know the nature of the hierarchy of masses, either Normal Hierarchy (2). In the NH case, for m ν 1 m ν 2 one gets approximately The lightest neutrino may be arbitrarily light from these data. In the case of the inverted hierarchy one has with m ν 3 arbitrarily light. Using these experimental data, we will constraint the possible values of the lightest neutrino in both NI and IH arising from the above WGC motivated constraint, the absence of stable AdS vacua 1 We discuss the cases of Majo-1 It has been recently claimed a slight preference for the natural hierarchy from the combined analysis of neutrino data [24][25][26]. However the evidence is still very weak [27]. rana and Dirac neutrinos in turn.

Majorana neutrinos
In the case of Majorana neutrinos we have 6 fermionic and 4 bosonic degrees of freedom, so one expects that AdS vacua will develop. In Fig. 3

the effective potential for
Majorana neutrinos is shown where the lightest neutrino has a zero mass. On the left panel of Fig. 3 it is assumed a NH for the neutrinos masses where on the right panel of Fig. 3 it is assumed an IH. An AdS vacuum is always formed for this configuration in both hierarchies. If the mass of the lightest neutrino is different from zero, the mass terms of the potential make the potential deeper. So the case of the pure SM with Majorana neutrino masses would be ruled out, as already advanced in [11].

NH IH
No vacuum m ν 1 < 6.7 meV m ν 3 < 2.1 meV dS 3 vacuum 6.7 meV < m ν 1 < 7.7 meV 2.1 meV < m ν 3 < 2.56 meV  AdS vacua and should be forbidden. We see that in the NI case this bound is around 7 times higher than the experimental Λ 4 and several orders of magnitude higher in the IH case. That is why in the Majorana case is impossible to avoid AdS vacua. We will see later however, that the addition of e.g. just an additional light Weyl fermion to the SM it is enough to make viable the Majorana neutrino case.
In the case of Dirac neutrinos the situation is different due to the fact that the  The form of the curves in figs. 5,6 may be understood from the aproximate equations given in 3.14. Let us assume for the moment that mR is small for the three neutrinos, so the formula 3.14 is a good approximation. We can then minimize the potential to get scales as m 4 ν , as observed experimentally. The mere existence of these lower bounds is interesting, since the only input is the value of the 4D c.c., yet the values obtained are close to expectations from particle physic models. Furthermore they give us a rationale as to why a non-vanishing value of the c.c. would be expected on arguments not based at all on the need for dark energy in cosmology. The existence of dark energy could have been predicted on the basis of these arguments.

AdS Casimir SM vacua in 2D
In the previous section it was shown the 3D compactification of the SM. One can also compactify to 2D [4,6]. In this case there are more SM compactifications than in the 3D case. The most simple case is the compactification in a 2D torus, and this is the case that we will follow in the rest of the work. However several compactifications in different manifolds are also possible. One example is the 2D sphere. In this case there is an extra classical contribution from the curvature to the potential. As it was shown in [6]

The radion potential in 2D. 2D SM vacua and neutrino masses
As pointed out in Ref. [4] and then studied in detail in Ref. [6] the 2D potential can be written as with V 2D−C [a, τ 1 , τ 2 , m a ] defined by where the 2D torus is parametrized as In the following we will assume |τ | = 1 for the torus.
The minima of the effective potential corresponding to AdS vacua are those verifying the conditions [4,6] V (a, τ ) = 0, These stationary points in the case of the 2D torus are τ = 1 and τ = 1/2 + i √ 3/2.
As it was pointed out in Ref. [6] only the latter point presents a minimum of the potential. Thus, in the rest of the paper we assume this structure for the 2D torus in the computations. It is important to note that in this case either an AdS vacuum appears or there is no vaccuum at all. This scenario is different compared to the 3D case since in the latter also dS vacua could appear.

Majorana neutrinos
As it was discussed in the case of the 3D compactification, we expect the presence of AdS vacua due to the fact that there is a bigger number of fermionic degrees of freedom compared to the bosonic ones. In Fig. 7 we show the potential for Majorana neutrinos. Left panel of Fig. 7 corresponds to a NH ordering and the right panel to an IH. As in the case of 3D compactification an AdS vacuum always develops. For a

Dirac neutrinos
In the case of Dirac neutrinos, one could expect the same possibilities that we found for the 3D case. In this 2D case however we can conclude immediately when an AdS vacuum is present since there are not dS vacua as it was mentioned before. In Fig. 8

Cosmological constant versus neutrino masses from 2D vacua
As we did in the 3D compactification case, we can compute how the 4D cosmological constant value could affect the creation of AdS vacua. In Fig. 9 the lower bound on the cosmological constant as a function of the lightest neutrino mass is depicted for the case of Majorana neutrinos. The red area corresponds to AdS vacua and so it is excluded. The left panel of Fig. 9 shows a NH for Majorana neutrinos and the right one for IH. In comparison with the 3D case we see that the limits are more stringent. For the NH scenario we have that the lower value for the cc to have a non-AdS vacuum is 60 times larger than the cc, while in the 3D case this number was 7. This is also de case for IH where now the limits on the minimal value of the cc are one order of magnitude larger. For the Dirac case something similar happens as we can deduce from

Adding Weyl, Dirac fermions or gravitino
The existence of very light neutral fermionic degrees of freedom have been advocated for several purposes. Some examples are as follows: Sterile neutrinos. These particles have been introduced as a generalization of the SM neutrino system (see e.g. [28] for reviews). One original motivation was the presence of such states with a mass of order 1eV to account for some neutrino oscillation anomalies detected at LSND and other neutrino experiments. But, more generally, the presence of sterile neutrinos has been considered for a variety of purposes. Axinos (SUSY partners of the axion) may also be considered in this class. Although in specific models sterile neutrinos have masses typically of order 1 eV, very light sterile neutrinos with masses e.g. m 2 = 6 × 10 −3 eV 2 , relevant for Casimir energies, are also possible (see e.g. [29]).
Light gravitinos. Very light gravitinos appear in models of low scale gauge mediation. Minimal models of gauge mediation have gravitino masses of order  Let us finally mention that ultralight fermionic states may contribute to the effective number of degrees of freedom N ef f in cosmology. But the limits apply to particles who were at some point in thermal equilibrium with the SM and decoupled before recombination. Details on bounds of dark radiation depend sensitively on how and when the particle decoupled and hence need not apply to the light degrees of freedom here considered (see e.g. [32,33]).

Independently of any motivation, it is clear that additional Weyl or Dirac fermions
with masses relevant for the Casimir potential could be present in addition to the SM from e.g. hidden sectors or dark portals. Here we will present results for the addition of one or two Weyl fermions. The case of two Weyl fermions yields the same as the addition of one Dirac fermion or a gravitino.  We have also worked out the same study for the case of 2D vacua. The results are shown in figs.17 and 18. They are almost identical to those we found for the 3D vacua.

One Weyl fermion
The limits are slightly stronger but one can barely note the difference.
As a summary, adding a Weyl or a Dirac fermion sufficiently light to Majorana neutrinos make the latter viable with the present constraints. The lightest Majorana neutrino would be amenable to planned ν-less double β-decay if we add a Dirac fermion or a gravitino.

Axions
Axion-like particles are natural candidates for BSM states populating the infrared sector of the SM. Their shift symmetry a → a + 2πf a protects their masses from quantum corrections and make ultralight masses natural. The best motivated such particle is the QCD axion which is introduced to solve the strong CP problem of QCD.
The mass of the QCD axion is given by (see e.g. [36] and references therein) In addition to the QCD axion, the existence of other axion-like particles (ALP) has been suggested for a variety of purposes. For these ALP's the mass can vary in a very wide range. A recently popular ALP is the relaxion [37] in which the minimal model has a mass as low as m a 10 −25 eV. In the formulation of relaxion in terms of 4-forms [38], the mass of the relaxion is given by m a = F 4 /f a , where F 4 (10 −3 eV ) 2 is the 4-form field strength. An ALP coupled to 4-forms (a hierarxion [39]) and the Higgs particle has also been recently suggested in order to construct a landscape of values for the Higgss mass. In this case the ALP mass varies in a range 10 −17 eV < m a < 10 3 eV .
Axions or ALP's may constitute the dark matter in the universe. Recently the case of ultralight scalars with mass m a 10 −22 eV constituting what is called fuzzy dark matter has been studied (see e.g. [40] and references therein). All these possible sources of axion-like particles could if present contribute to the potential of the radion. The axion contribution to the general effective potential would be negative due to its bosonic nature. In principle the axion contribution to the 3D potential reads, 1 n 2 K 2 (2πRm a n).

(5.4)
However, besides the Casimir contribution to the potential, there is an extra contribution to the potential from axionic fluxes [4] . The field strength of tha axion (da) can be non-vanishing around the compact circle S 1 2 , with the flux f quantized as f = 2πnf a , with f a the axion periodicity (decay constant).
This flux contributes to the effective potential a piece so that the full axion contribution to the potential is given by Since the value of f a is enormously large in specific ALP's, the flux contribution completely overwhelms the Casimir contribution for non-vanishing fluxes n = 0. This depends on its number so we will distinguish two cases.

One axion
Let us consider first the case of one single axion 3 . In the case of Majorana neutrinos the addition of an axion does not change things. Since an axion contributes negatively, an AdS vacuum still develops and becomes in fact deeper, since there are 6 fermionic degrees of freedom and 5 bosonic.
In the case of Dirac neutrinos, the negative contribution of the axion slightly modifies some of the vacua, some of them could also change its nature or even create new vacua which were absent in the axion-less case. The results depend on the relative magnitude of the axion mass and the mass of the heaviest neutrinos as well as whether the neutrino hierarchy is normal or inverted. In Fig. 19 the effect of the axion for the different vacua formation is shown in the lightest neutrino mass and axion mass plane.
For this plot we have assumed that n = 0 so there is no contribution from the flux term. We have analysed masses of the axion from 10 −6 eV to 10 5 eV. We can see the different effects that an axion could produce for NH and IH hierarchies: • NH Dirac neutrinos In the NH case we see that for axion masses above around 10 −2 eV the number of light states becomes the same as in the axion-less case, and we recover the limit m ν 1 < 7.7 × 10 −3 eV. For axions lighter than 10 −2 eV the effective number of degrees of freedom decreases one unit and the bound becomes stronger, m ν 1 <

meV.
• IH Dirac neutrinos For IH, when we include an axion field an AdS vacuum is created even when the lightest neutrino mass is set to zero, m ν 3 = 0 eV. The reason for this behaviour is the fact that in IH there are two heavy states that even when the lightest neutrino mass is set to zero their masses are m ν 1 ∼ m ν 2 ∼ 50 meV. In this case there are 5 bosonic degrees of freedom against 4 fermionic ones below 50 meV, so an AdS vacuum is formed. Note that the QCD region of axion masses would then be excluded for Dirac neutrino masses, which is a strong result. On the other hand, when the axion mass reaches the heavy neutrino states masses the fermionic degrees of freedom start contributing to the effective potential. In that sense, when the mass of the lightest neutrino is set to zero, m ν 3 = 0 eV, one

Multiple axions and axiverse
For more than one axion-like particle the situation may change in an important way.
The reason is that if a sufficiently large number of axions have their masses in the for NI (IH) respectively. If the lightest neutrino Majorana mass is induced from a standard see-saw mechanism one obtains (e.g for NI) 5 where for NI(IH). Now, for fixed Yukawa coupling the EW scale is again bounded above by the 4D cosmological constant. In the Dirac case, though, the Yukawa coupling needs to be extremely small to match the scale of observed neutrino masses 6 . But again, the smallness of the cosmological constant implies in turn a small EW scale in order to be consistent with quantum gravity. This relation is shown on the right panel of figure 21 for fixed Yukawa coupling Y = 10 −14 .   One first interesting result is the existence of a lower bound on the value of the c.c.
in terms of the light degrees of freedom of the SM Casimir potential. One can obtain an approximate analytic expression of the form This is interesting because, as far as we are aware, this is the only known suggestion for a non-vanishing value of the c.c. related to neutrino masses and independent of any cosmological argument (dark energy). Before evidence for an accelerating universe was found, it was widely believed that Λ 4 = 0. The conjecture here studied would have implied the existence of a 4D c.c. to avoid inconsistent AdS vacua, independently of any cosmological argument.
We find that the existence or not of dangerous lower dimensional SM AdS vacua is very sensitive both to the value of Λ 4 , neutrino masses and possible BSM extensions.
We have done a systematic study of this dependence for both 3D and 2D SM vacua and the summary is shown in table 5. The results for 2D and 3D vacua are quite similar, although bounds coming from the absence of 2D vacua are slightly stronger.  Table 5: Conditions on neutrino, fermion and axion masses (in eV) from the absence of 3D and 2D SM vacua. Here yes means that no AdS value forms independently of the values of parameters, no means the opposite. Note that the 2D constraints are slightly stronger than the 3D constraints but follow a similar patern. Majorana neutrino masses accessible to ν-less double β-decay require the existence of either at least 2 additional weyl spinors or 2 or more scalars (e.g. axions or ultralight vector bosons).
Perhaps the most attractive setting for neutrino masses is that of Majorana neutrinos (from a see-saw mechanism) in normal hierarchy. If no additional BSM states are around, Majorana neutrinos are not consistent with the bounds from absence of AdS vacua here discussed, as already pointed out in [11]. However we have found that An interesting light addition to the SM is that of an axion. If only one axion is added, Majorana neutrinos still lead to undesired AdS vacua and would be ruled out.
In the case of Dirac neutrinos absence of dangerous vacua are obtained if the lightest neutrinos have m ν 1 ≤ 4 × 10 −3 eV for NI (m ν 3 ≤ 1 × 10 −3 eV for IH). In this latter case however the axion must have m a ≥ 2 × 10 −2 eV, so that it cannot be identified with a standard QCD axion.
The existence of 3D,2D SM vacua can leave an imprint in cosmology (see [42]).
Indeed if our universe came from a lower dimensional one in 2+1 dimensions there could be some detectable imprints, due to the anisotropy of space. This may affect the CMB if the last period of inflation was not too long. This effect would appear at the highest multipoles. However we have just seen that AdS SM vacua cannot be stable, so that such anisotropies could not originate from such primordial vacua. Only dS 3D vacua would still be possible, but we have seen that such vacua only appears for very narrow regions for the mass of the lightest neutrino. Thus e.g. in the case of Dirac neutrinos 3D dS vacua only appear in the region 6.7 × 10 −3 eV ≤ m ν 1 ≤ 7.7 × 10 −3 eV for NI or 2.1 × 10 −3 eV ≤ m ν 3 ≤ 2.56 × 10 −3 eV (see Table 1 We find quite remarkable that a very abstract condition like the absence of stable AdS vacua may give rise to such a wealth of implications on magnitudes of direct physical relevance like the cosmological constant, neutrino masses and even the origin of the EW hierarchy. In overall, our analysis is a clear example of how consistency with quantum gravity can have important implications on IR physics. Not all points in the parameter space leading to different quantum field theories are allowed when including gravity, and apparent fine-tuning problems can turn out to be only mirages due to our ignorance of the actual landscape of consistent theories. This can force us to review our notions of naturalness and the hierarchy problems in particle physics when combined with quantum gravity.