Searching for neutrino-modulino oscillations at the Forward Physics Facility

We make use of swampland conjectures to explore the phenomenology of neutrino-modulino mixing in regions of the parameter space that are within the sensitivity of experiments at the CERN’s Forward Physics Facility (FPF). We adopt the working assumption of Dirac mass terms which couple left-and right-handed neutrinos. We further assume that the 3 right-handed neutrinos are 0-modes of bulk 5-dimensional states in the dark dimension, a novel scenario which has a compact space with characteristic length-scale in the micron range that produces a natural suppression of the 4-dimensional Yukawa couplings, yielding naturally light Dirac neutrinos. We formulate a specific realization of models with high-scale supersymmetry breaking that can host a rather heavy gravitino ( m 3 / 2 ∼ 250 TeV) and a modulino with mass scale ( m 4 ∼ 50 eV) within the FPF discovery reach. Neutrino oscillations imply

We make use of swampland conjectures to explore the phenomenology of neutrino-modulino mixing in regions of the parameter space that are within the sensitivity of experiments at the CERN's Forward Physics Facility (FPF).We adopt the working assumption of Dirac mass terms which couple left-and right-handed neutrinos.We further assume that the 3 right-handed neutrinos are 0-modes of bulk 5-dimensional states in the dark dimension, a novel scenario which has a compact space with characteristic length-scale in the micron range that produces a natural suppression of the 4-dimensional Yukawa couplings, yielding naturally light Dirac neutrinos.We formulate a specific realization of models with high-scale supersymmetry breaking that can host a rather heavy gravitino (m 3/2 ∼ 250 TeV) and a modulino with mass scale (m4 ∼ 50 eV) within the FPF discovery reach.
Neutrino oscillations imply the existence of new states that can generate neutrino mass terms consistent with the Standard Model (SM) SU (2) gauge symmetry.Since the observed neutrino mass splittings are tiny, a compelling realization comes from string models with large extra dimensions [1], in which gravitons and right-handed neutrinos are allowed to propagate in the bulk, whereas the SM fields are confined to localized 3-branes.Within this set up the 4-dimensional neutrino Yukawa couplings are suppressed relative to charged-fermion Yukawa couplings by a factor proportional to the square root of the volume of the extra dimensions [2][3][4][5][6].In other words, neutrinos are very light for the same reason that gravity appears to be very weakly coupled.
In addition, it has long been suspected that Planck (M p ≃ 2.4 × 10 18 GeV) suppressed interactions could be connected to neutrino physics, because the coupling M −1 p LLHH generates a mass ⟨H⟩ 2 /M p ∼ 10 −1 meV near the neutrino mass scale, where L is the leptonic doublet and H the Higgs doublet of the SM [7,8].It has also been suspected that in superstring theory the superpartners of moduli fields provide compelling candidates for the right-handed neutrinos [9,10].Before supersymmetry (SUSY) breaking, some of these moduli fields as well as their fermionic partners are exactly massless.Masses for the moduli and modulinos are then generated when SUSY is broken and can be small.This may account for the lightness of any sterile (s) neutrino states.
In this Letter we reexamine these captivating ideas within the context of the swampland program [11], focussing attention on neutrino-modulino oscillations which can be probed by experiments at the CERN's Forward Physics Facility (FPF) [12,13].We formulate a spe-cific scenario of high-scale SUSY breaking that can host a rather heavy gravitino and a modulino with mass scale within the FPF discovery reach.In a companion paper we discuss theoretical aspects of the model and explore in more detail the phenomenology of neutrino-modulino mixing in models with high-and low-scale SUSY breaking [14].
The swampland program keeps within bounds the set of 4-dimensional effective field theories (EFTs) that are a low energy limit of quantum gravity, and differentiate these theories from those that are not.The plan of action is accomplished by conjecturing guiding principles that any EFT should satisfy in order to be in the landscape of supersting theory vacua, rather than be relegated to the swampland [15][16][17].Recently, by putting together predictions from the swampland program with observational data it was elucidated that the smallness of the cosmological constant in Planck units (Λ ∼ 10 −120 M 4 p ) seems to indicate that our universe could stretch off in an asymptotic region of the string landscape of vacua [18].
More concretely, the distance conjecture predicts the emergence of infinite towers of Kaluza-Klein (KK) modes that become exponentially light in Planck units (M p = 1), yielding a breakdown of any EFT at infinite distance limits in moduli space [19].The related anti-de Sitter (AdS) distance conjecture correlates the dark energy density to the mass scale of the infinite tower of states [20].Now, if the AdS distance conjecture is to be generalized to dS space, the dark energy scale Λ can be accommodated with the addition a mesoscopic (dark) extradimension characterized by a length-scale in the micron range.
All in all, in this realization of a universe with tiny arXiv:2308.11476v3[hep-ph] 11 Feb 2024 vacuum energy a dark dimension opens up at the characteristic mass scale of the KK tower, where 10 −2 ≲ λ ≲ 10 −4 [18].The 5-dimensional Planck scale (or species scale where gravity becomes strong) is given by where N is the number of the quantum field species below Λ QG [21,22], here to be identified with the number of KK modes.
In the spirit of [2][3][4][5][6], we assume the generation of neutrino masses originates in 5-dimensional bulk-brane interactions of the form where H = −iσ 2 H * , L i denotes the lepton doublets (localized on the SM brane), Ψ j stands for the 3 bulk (righthanded) R-neutrinos evaluated at the position of the SM brane, y = 0 in the fifth-dimension coordinate y, and h ij are coupling constants.This gives a coupling with the L-neutrinos of the form ⟨H⟩ ν Li Ψ j (y = 0), where ⟨H⟩ = 175 GeV is the Higgs vacuum expectation value.Expanding Ψ j into modes canonically normalized leads for each of them to a Yukawa 3 × 3 matrix suppressed by the square root of the volume of the bulk where R ⊥ ∼ m −1 KK1 is the size of the dark dimension and M s ≲ Λ QG the string scale, and where in the second rendition we have dropped factors of π's and of the string coupling.
Note that KK modes of the 5-dimensional right-handed neutrino fields behave as an infinite tower of sterile neutrinos, with masses proportional to m KK1 .However, only the lower mass states of the tower mix with the active SM neutrinos in a pertinent fashion.The non-observation of neutrino disappearance from oscillations into sterile neutrinos at long-and short-baseline experiments places a 90% CL upper limit on the size of the dark dimension: R ⊥ < 0.4 µm for the normal neutrino ordering, and R ⊥ < 0.2 µm for the inverted neutrino ordering [24,25]. 1  This set of parameters corresponds to λ ≲ 10 −3 and so m KK1 ≳ 2.5 eV [28].
Before proceeding, it is important to stress that the upper bounds on R ⊥ discussed in the previous paragraph are sensitive to assumptions of the 5 th dimension geometry.Moreover, in the presence of bulk masses, the mixing of the first KK modes to active neutrinos can be suppressed, and therefore the aforementioned bounds on R ⊥ can be avoided [26,27].It is also worth mentioning that such bulk masses have the potential to increase the relative importance of the higher KK modes, yielding distinct oscillation signatures via neutrino disappearance/appearance effects.We will discuss this multiparameter scenario in [14], herein we focus on the simplest one-parameter model unescorted by bulk masses.
Without further ado we bring into play the modulino.Following [29], we assume that among the modulinos there is at least one, s, with the following properties: • s has only Planck mass suppressed interactions with SM fields, as expected for geometrical moduli governing the different couplings between light fields.We work within a simple construct, in which the relevant light scale for SM singlets is the gravitino mass m 3/2 and a dimensionless coupling constant with visible matter given by where α i = O(1).• The mixing of s with the active SM neutrinos involves the electroweak symmetry breaking, and the simplest appropriate effective operator is λ i L i sH.This operator generates a neutrino mass term m νis νi s with • The mass of the modulino, m 4 , is induced via SUSY breaking.We assume that m 4 is absent at the level m 3/2 and appears as where β = O(1).In [14], we will discuss these assumptions and the different realizations in details.Here, instead we will focus on the phenomenological implications.
To develop some sense for the orders of magnitude involved we make contact with experiment.The FPF is a proposal to build a new underground cavern at the Large Hadron Collider (LHC) to host a suite of far-forward experiments during the high-luminosity era [12,13].The existing large LHC detectors have un-instrumented regions along the beam line, and so miss the physics opportunities provided by the enormous flux of particles produced in the far-forward direction.In particular, the FPF proof of concept FASER has recently observed the first neutrinos from LHC collisions [30].During the high-luminosity era, LHC collisions will provide an enormous flux of neutrinos originating from the decay of light hadrons that can be used to probe neutrino-modulino mixing.We particularize our calculations to the design specifications of the FLArE detector, which will have a 1 m × 1 m cross sectional area with a 10 ton target mass, and it will be located at about 620 m from the ATLAS interaction point.
Considering typical energies O(TeV) of LHC neutrinos and a baseline of L ∼ 620 m, FLArE will be sensitive to modulino-neutrino square mass difference satisfying ∆m 2  41 which implies ∆m 2 41 ∼ 2000 eV 2 [31].It turns out that if m 3/2 ∼ 250 TeV, substituting for β ∼ 1.7 into (7) we obtain the required m 4 .We note in passing that the abundance of unstable gravitinos with m 3/2 ≲ 10 TeV is severely constrained by the success of the big-bang nucleosynthesis [32].Now, we want to investigate whether SUSY models hosting 250 TeV gravitinos live in the string landscape or the swampland.The first step towards associating m 3/2 to the mass scale of an infinite tower of KK modes was taken in [33], and this idea has been recently formulated as the gravitino conjecture [34,35].The dark dimension and the gravitino conjecture give rise to two possible schemes, depending on the relation between the corresponding towers of external states [36].A first possibility is that Λ and m 3/2 are connected to the same KK tower.A second possibility is that the towers are different.
In the single-tower scenario, the main formula of the gravitino conjecture leads to where n and λ 3/2 are swampland parameters [36].Note that for fiducial values λ ∼ 10 −4 , λ 3/2 ∼ 1, and n = 2 we obtain m 3/2 ∼ 250 TeV.This set of parameters lead to a high-scale SUSY breaking where we have taken κ ∼ 10 3 to accommodate the requirement M SUSY ≲ Λ QG ∼ 2.6 × 10 10 GeV.Alternatively, we can assume that together with the infinite tower associated to the dark dimension there is a second tower of KK states related to p additional compact dimensions, with mass scale m KK2 .In this case, the quantum gravity cut-off is given by [36] and the gravitino conjecture yields We select as fiducial parameters λ ∼ 10 −4 , n = 1, p = 1, m KK2 ∼ 10 9 GeV, and λ 3/2 ∼ 2.5 × 10 −4 .Our choice leads to m 3/2 ∼ 250 TeV and Λ QG ∼ 1.4×10 10 GeV.For the two-tower scheme the SUSY breaking scale satisfies, and so for κ ≳ 10 3.6 we have M SUSY ≲ Λ QG .Let's now provide a concise overview of concrete realizations [14].In linear supergravity realization, the natural size expected for the gravitino mass is of order of the compactification scale.A brane-localized gravitino mass term act as deformation of the gravitino wave function in the bulk and simply result in shifts in the KK mass spectrum by fractions of ∼ 1/R ⊥ .Working with only one flat extra-dimension leads therefore to gravitino mass ∼ 1/R ⊥ in the eV range, too small for our purpose.We instead contemplate the breaking of supersymmetry through non-periodic boundary conditions or fluxes in the other extra dimensions, anyway always present.To achieve m 3/2 ∼ λ 3/2 m KK2 ∼ 250 TeV while employing a dimension of size approximately M −1 s , we must introduce a small value for λ 3/2 .Within our framework, we can indeed consider the viability of a value around λ 3/2 ∼ 10 −4 .However, in general questions regarding the compatibility of such a small parameter with the gravitino conjecture.Adopting λ 3/2 = 1/2, we need to consider this dimension to be significantly larger than the string scale, as Λ QG ∼ 2 × 10 9 GeV and m KK2 ∼ 5 × 10 5 GeV.
Alternatively, we can consider p > 1 and invoke supersymmetry breaking via fluxes, leading to Then, by comparing with (12) this corresponds to the choice n = (p + 3)/4p and . By setting p = 3 and λ3/2 = 1/2, we obtain m 3/2 ∼ 250 TeV.It addition, from (11) it follows that M s ∼ Λ QG ∼ 3 × 10 9 GeV and also that m KK2 ∼ 2.6 × 10 8 GeV.For a particular example, we note that the modulino could be the fermionic partner of the radion. 2In what follows, we concentrate on this possible two-tower scheme and study the effects of neutrino-modulino oscillations.
Before proceeding, we pause to note that the sharpened version of the weak gravity conjecture forbids the existence of non-SUSY AdS vacua supported by fluxes in a consistent quantum gravity theory [37].Actually, the so-called non-SUSY AdS conjecture constrains the mass spectrum of the EFT in the far-infrared region, because the existence of AdS vacua depends on the neutrino masses and possible additional very light degrees of freedom via the Casimir potential [38].In particular, the simplest EFT with 3 Majorana neutrinos would be rule out, because when the SM coupled to gravity is compactified down to three dimensions AdS vacua appear for any values of neutrino masses consistent with experiment [39].For the model at hand, the Dirac neutrino states and associated KK modes prevent the rise of AdS vacua [27].
Next, in line with our stated plan, we show that neutrino-modulino oscillation effects can be probed by neutrinos produced at the LHC [40].Neutrino-modulino oscillations are analogous to those in the 3+1 neutrino mixing framework, in which the flavor states are given by the superposition of four massive neutrino states, where α = e, µ, τ, s [41].The presence of the modulino reshapes the active neutrino mixing angles via the unitarity relations of the mixing matrix U, i.e., α U * αi U αj = δ ij and where Greek indices run over the neutrino flavors and Roman indices over the mass states.Bearing this in mind, the unitary 4 × 4 mixing matrix takes the form U = U 34 U 24 U 14 U PMNS where U PMNS = U 23 U 13 U 12 is the the Pontecorvo-Maki-Nagakawa-Sakata (PMNS) matrix [42][43][44].
Neutrino-modulino oscillations are driven by the Hamiltonian where ) and E is the neutrino energy.The ν α → ν β transition probability is 2 In the standard moduli stabilization by fluxes, all complex structure moduli and the dilaton are stabilized in a supersymmetric way while Kahler class moduli need an input from SUSY breaking.The radion is Kahler class and exists in a model independent fashion within the framework of the dark dimension.found to be where L is the experiment baseline.In the FPF-shortbaseline limit (viz., ∆m 2 21 L/E ≪ 1 and ∆m 2  31 L/E ≪ 1) for which SM oscillations have not developed yet, the effective oscillation probabilities can be written as Using (16), the transition probability (19) can be recast as where sin In Fig. 1 we show a comparison of the SM neutrino oscillation probability and the neutrino-modulino oscillation probability for the disappearance channel assuming representative parameters.We can see that using future FLArE data we will be able to constrain the neutrinomodulino mixing angle by comparing the high-energy bins, where there is no oscillation of the SM neutrinos, to the bins at the neutrino-modulino oscillation minimum (around 1 TeV) where there is a deficit.
As an illustration, Fig. 2 displays the effect of electron neutrino disappearance due to a modulino of mass m 4 = 50 eV with |U e4 | 2 = 0.1.It is clearly seen that the expected flux for the disappearance channel can be distinguished from the prediction of the SM three flavor scheme, demonstrating that the FPF could test the neutrino-modulino parameter space.We end with a discussion of constraints on sterile neutrinos from astrophysical and cosmological observations, as well as searches in collider and beam dump experiments.
• At the intensity frontier sterile neutrinos can be produced abundantly via meson decay in beamdump experiments.These experiments, however, probe very heavy sterile neutrinos with masses in the range MeV ≲ m 4 ≲ GeV [46][47][48][49].• Active-sterile neutrino mixing is strongly constrained for m 4 ≳ 100 keV to avoid excessive energy losses from supernova cores, but in the range m 4 ≲ 10 keV the mixing angle is essentially unconstrained [50].• Big bang nucleosynthesis can also probe properties of sterile neutrinos.However, the 2 H and 4 He abundances are mostly sensitive to the sterile neutrino lifetime and depends only weakly to the way the active-sterile mixing is distributed between flavors [51].• Active neutrinos are produced through β-decays of unstable isotopes and in nuclear fission processes.Sterile neutrinos can also be produced via the active-sterile mixing if the sterile mass is smaller than the energy release of the relevant nuclear process.In this direction, the β-decay spectra of 18 F, 19 Ne, 35 S, and 45 Ca probe the range 4 < m 4 /keV < 2000 [52][53][54][55][56], whereas the spectra of the 187 Re and ], saturating the mixing strength adopted for illustrative purposes in Fig. 2. KATRIN Collaboration reported the most restrictive bound from the tritium β-decay spectrum in the mass range 0.1 < m 4 /keV < 1.0 [60].All in all, we conclude that there is a large window of sterile mixing in the modulino mass range to be probed by experiments at the FPF.A thorough study of the sensitivity of FPF experiments, including systematic uncertainties, is beyond the scope of this paper.Actually, we defer this type of study to the FPF Collaboration; first steps are underway [61].
In summary, we have put together a scenario with high-scale SUSY breaking that can host a rather heavy gravitino and a modulino with a mass of about 50 eV.The corresponding models are fully predictive through neutrino-modulino oscillations and can be confronted with data to be collected by the FPF experiments.

Saturday, August 5, 23 FIG. 2 :
FIG. 2: The expected energy spectrum of interacting electron neutrinos in the FLArE detector is shown as the solid gray line.The corresponding statistical uncertainties are shown as black error bars.The colored dashed line shows how neutrino-modulino oscillations would change the expected flux if the disappearance probability were characterized by ∆m 2 41 = 2500 eV 2 and |Ue4| 2 = 0.1.The simulations have been normalized assuming an integrated luminosity of 3 ab −1 and using the design specifications of the FLArE detector, which is supposed to have a 1 m × 1 m cross sectional area, a 10 ton target mass, and L = 620 m.Adapted from[45].