Dark Neutrinos and a Three Portal Connection to the Standard Model

We introduce a dark neutrino sector which respects a hidden $U(1)^\prime$ gauge symmetry, subsequently broken by the vacuum expectation value of a dark scalar. Generically, this hidden sector communicates with the SM only via the three renormalizable portals, namely neutrino, vector and scalar mixing. We highlight the fact that in this unified picture the phenomenology can be significantly different from that of each individual portal taken separately. Several bounds become much weaker or can be avoided altogether. Novel signatures arise in heavy neutrino, dark photon and dark scalar searches, typically characterised by multi-leptons plus missing energy and displaced vertices. A minimal extension, possibly motivated by anomaly cancellations, can accommodate a dark matter candidate, strongly connected to the neutrino sector.


INTRODUCTION
The two most important evidences that the Standard Model (SM) of particle physics is not complete are the existence of neutrino masses and mixing, and dark matter (DM) in the Universe. Both call for extensions of the SM and the possible existence of dark sectors which do not partake in SM gauge interactions, or do so with extremely weak couplings while displaying strong "dark" interactions [1,2]. Such sectors might exist at relatively light scales below the electroweak one, being within reach of present and future non-collider experiments. Generically, a neutral dark sector can communicate with the SM via three renormalizable portals. New neutral fermions mix with light neutrinos unless a symmetry differentiates the two, a possibility usually denoted as neutrino portal. The vector portal comes from the kinetic mixing between a hidden Z ′ and the SM Z and/or photon. This term is generically allowed in the Lagrangian and an explanation of its smallness requires specific UV completions. Similarly, in presence of a scalar acquiring a vacuum expectation value, the scalar portal arises due to the mixing with the Higgs boson.
In this article, we propose a new neutrino model with a hidden U (1) ′ gauge symmetry under which no SM fields are charged. We introduce new SM-neutral fermions, ν D and an additional sterile neutrino N . The symmetry is subsequently broken by the vacuum expectation value (vev) of a complex dark scalar Φ, which gives mass to the new gauge boson. For concreteness, we restrict the scale of the breaking to be below the electroweak one. Models with heavy neutrinos which are not completely sterile and might participate in new gauge interactions have been studied in several contexts, including B −L, L µ −L τ and left-right symmetric models [3][4][5][6][7][8][9][10][11], but here we focus on the possibility of a symmetry under which no SM fields are charged [12][13][14]. New heavy neutral fermions that feel such hidden forces, such as ν D , are referred to as dark neutrinos, since they define a dark sector separate from the SM. Nevertheless, the dark interactions "leak" into the SM sector via neutrino mixing, where they may dominate [15,16]. Models of this type have been invoked to generate large neutrino non-standard interactions [17,18], generate new signals in DM experiments [15,[19][20][21], weaken cosmological and terrestrial bounds on eV scale sterile neutrinos [22,23], and as a potential explanation of anomalous short-baseline results at the MiniBooNE [24] and/or LSND [25] experiments with new degrees of freedom at the MeV/GeV scale [26][27][28][29][30][31][32].
Our model presents all the three renormalizable portals to the SM. The Yukawa interactions between the leptonic doublet and N , and between N and ν D induce neutrino mixing. The gauge symmetry allows a crosscoupling term in the potential between the Higgs and the real part of the scalar, inducing mixing between the two after symmetry breaking. The broken gauge symmetry implies the existence of a light hidden gauge boson X µ , which mediates the dark neutrino interactions and generically kinetically mixes with the SM hypercharge. The set-up is self-consistent and combines the three portals into a unified picture that exhibits significantly different phenomenology with respect to each portal taken separately, as we discuss. The interplay of the different portal degrees of freedom leads to novel signatures which would have escaped searches performed to date, and that can even explain long-standing anomalies. For the latter, we focus on the MiniBooNE anomaly as dis- cussed in Ref. [30] (see also [31]) and on new neutrino scattering signatures at neutrino experiments [32]. We also reconsider the possibility to explain the discrepancy between the prediction and measurement of the anomalous magnetic moment of the muon (∆a µ ) [33] via kinetic mixing [34].
An interesting feature of the model is the generation of neutrino masses at loop-level. This requires only two key features of our setup, namely a light Z ′ and neutrino mixing, but not the vector and scalar portals. For this reason, we discuss it elsewhere [35].
In its minimal form, the model is not anomaly-free. We discuss how this can be cured and propose a minor extension that introduces additional dark sector neutral fermions charged under the new symmetry [1]. Neutrinos, we argue, may be a window into such dark sectors, bridging the puzzles of neutrino masses and DM [36][37][38][39][40][41][42][43][44][45][46]. We briefly outline the key features of a DM extension and leave a more detailed analysis to future work.

THE MODEL
We extend the SM gauge group with a new abelian gauge symmetry U (1) ′ with associated mediator X µ and introduce three new singlets of the SM gauge group: a complex scalar Φ, and two left-handed fermions ν D,L ≡ ν D and N L ≡ N . The scalar Φ and the fermion ν D are equally charged under the new symmetry, and N is neutral with respect to all gauge symmetries of the model. For simplicity, we restrict our discussion to a single generation of hidden fermions. The relevant terms in the gauge-invariant Lagrangian are where X µν is the field strength tensor for X µ , D µ ≡ (∂ µ − ig ′ X µ ) the covariant derivative, L α ≡ (ν T α , ℓ T α ) T the SM leptonic doublet of flavour α = e, µ, τ and H ≡ iσ 2 H * is the charge conjugate of the SM Higgs doublet. We write y α ν for the L α -N Yukawa coupling, y N for the ν D -N one, and µ ′ for the Majorana mass of N , which is allowed by the SM and the new gauge interaction, although it breaks lepton number by 2 units.
The minimisation of the scalar potential V (Φ, H) leads the neutral component of the fields H and Φ to acquire vevs v H and v ϕ , respectively. The latter also generates a mass for both the new gauge boson X µ and the real component of the scalar field ϕ. Although v ϕ is arbitrary, we choose it to be below the electroweak scale, v ϕ < v H , as we are interested in building a model testable at low scales.
The interactions of the dark neutrino with the SM arise due to the so-called portal couplings, shown in Fig. 1. We discuss these in detail now.
Neutrino portal In the neutral fermion sector and after symmetry breaking, two Dirac mass terms are in- It is useful to consider the form of the neutrino mass matrix in the single generation case to clarify its main features. For one active neutrino ν α (α = e, µ, τ ), it reads The form of this matrix appears in Inverse Seesaw (ISS) [47] and in Extended Seesaw (ESS) [48] models. In fact, it is the same matrix discussed in the so-called Minimal ISS [49], with the difference that in our case its structure is a consequence of the hidden symmetry. After diagonalisation of the mass matrix, the two heavy neutrinos, ν h with h = 4, 5, acquire masses. Assuming that m D ≪ Λ, we focus on two interesting limiting cases. In the ISS-like limit, where Λ ≫ µ ′ and the two heavy neutrinos are nearly degenerate, we have In the ESS-like case, Λ ≪ µ ′ , one neutral lepton remains very heavy and mainly in the completely neutral direction N , and the other acquires a small mass via the seesaw mechanism in the hidden sector. We find From the discussion above, it is clear that the masses of Z ′ and ϕ ′ are typically above the heavy neutrino ones, unless we are in the ESS-like regime. The Yukawa terms in Eq. (1) induce neutrino mixing between the active (light) and heavy (sterile, dark) neutrinos. In this model, similarly to the ISS and the ESS cases, this mixing can be much larger than the typical values required in type-I seesaw extensions to explain neutrino masses, making its phenomenology more interesting. The determinant of the mass matrix in Eq. (2) is zero, and so light neutrino masses vanish at tree-level and do not constrain the values of the active-heavy mixing angles. This, however, is no longer the case at one-loop level, as light neutrino masses emerge through radiative corrections from diagrams involving the ϕ ′ and Z ′ degrees of freedom [35]. Scalar portal In the scalar potential, the symmetries of the model allow us to write down the following term where we identify λ ΦH as the scalar portal coupling [50], responsible for mixing in the neutral scalar sector. If such a term exists, the scalar mass eigenstates (h ′ , ϕ ′ ) mix with the gauge eigenstates (h, ϕ) with a mixing angle α defined by where λ h and λ ϕ are the quartic couplings of the Higgs and Φ scalars, respectively. Vector portal Similarly, mixing also arises in the neutral vector boson sector from the allowed kinetic mixing term [51] where F µν is the SM hypercharge field strength. This term may be removed with a field redefinition, resulting in three mass eigenstates A, Z 0 , Z ′ , corresponding to the photon, Z 0 -boson and the hypothetical Z ′ -boson. For a light Z ′ , the Z ′ coupling to SM fermions f to first order in the small parameter χ is given by with q f the fermion electric charge. The values of χ and λ ΦH are arbitrary and could be expected to be rather large. As such, we treat them as free parameters within their allowed ranges. Here, we merely note that with our current minimal matter content, χ and λ ΦH receive contributions at loop level from the (L α · H)N c and N ν c D Φ terms, which are necessarily suppressed by neutrino mixing (χ ∝ g ′ e|U αh | 2 and λ ΦH ∝ |U αh | 2 ). These values constitute a lower bound and larger values should be expected in a complete model.

Portal phenomenology
The interplay between portal couplings and the heavy neutrinos ν h (h = 4, 5) leads to a distinct, and possibly richer, phenomenology to what is commonly discussed in the presence of a single portal. We present here some of the most relevant signatures, devolving a longer study to future work.
Heavy neutrino searches The strongest bounds on Heavy neutrinos in the MeV-GeV mass range come from peak searches in meson decays [52][53][54] and beam dump experiments [55][56][57][58][59][60] looking for visible ν h decays. These, however, can be weakened if the ν h decays are sufficiently different from the case of "standard" sterile neutrinos with SM interactions suppressed by neutrino mixing. We now discuss how this may happen, depending on the mass hierarchy of the two heavy neutrinos and the values of neutrino and kinetic mixing. For concreteness, we focus on specific benchmark points (BP) that illustrate the key features. In the ISS-like regime, we take m 4 /m 5 = 99% and choose m 4 ≃ m 5 = 100 MeV. If χ is negligible, we have that ν h decays as in the standard sterile case via SM interactions. This is because the ν 5 → ν 4να ν α decay is phase-space suppressed (Γ ν5→ν4νν ∝ µ ′ 5 ), and because Z ′ mediated decays into three light neutrinos are negligible for small mixing, as Γ ν h →ννν ∝ |U αh | 6 m 5 h /m 4 Z ′ . If χ is sizeable, on the other hand, new visible decay channels dominate, specifically ν 4 → ν α e + e − for this BP. The corresponding decay rate is given by Depending on the value of χ and m ′ Z this decay can be much faster than in the SM, implying stronger constraints on the neutrino mixing parameters as discussed in Ref. [61]. For heavier masses, additional decay channels, e.g. ν 4 → ν α µ + µ − , would open. A feature of the model is that such channel would have the same BR as the electron one, albeit phase space suppressed. No twobody decays into neutral pseudoscalars arise due to the vector nature of the gauge coupling, unless mass mixing is introduced (see [62] for a thorough discussion of the decay products of a dark photon). We consider also a BP in the ESS-like regime. We take m 4 = m 5 /10. In this case, ν 5 decays into 3 ν 4 states very rapidly. The subsequent decays of ν 4 would proceed as discussed above and would be much slower than the ν 5 one, given the hierarchy of masses and the further suppression due to neutrino and/or kinetic mixing.
For large χ, peak searches and bounds on lepton number violation (LNV) from meson and tau decays may be affected [63,64]. Despite simply relying on kinematics, we note that in peak searches the strict requirement of a single charged track in the detector [53] would, in fact, veto a large fraction of new physics events if ν h decays promptly into ν α e + e − , for instance. In addition, LNV meson and tau decays would need to be reconsidered as the intermediate on-shell ν h could decay dominantly via the novel NC interactions and the ℓπ and ℓK final states would be absent.
Dark photon searches Bounds on the vector portal come from several different sources [65,66]. Electroweak precision data and contributions to the g − 2 of the muon and electron remaining valid in our model [67]. A major effort at collider and beam dump experiments has led to strong constraints on dark photons by searching for the production and decay of these particles. Such bounds, however, depend on the lifetime of the Z ′ and on its branching ratio (BR) into charged particles. In our model, the Z ′ can predominantly decay invisibly into heavy fermions if m Z ′ > 2m 4 and into light neutrinos otherwise. In the latter case, constraints would be much weaker than usually quoted with only mono-photon searches [68] applying. In the former case, however, new signatures arise, where the subsequent decay of ν h leads to multi-lepton/multi-meson events, potentially with displaced vertices and providing a very clean experimental signature. Notably, if the Z ′ decays into ν h states that subsequently decay sufficiently fast within the detector, even the "invisible decay" bounds will be weakened.
Revisiting a µ The above possibility opens the option to explain the discrepancy between the theoretical prediction [69] and the experimental value [33] of the (g − 2) of the muon via kinetic mixing. For instance, a 1 GeV Z ′ with χ = 2.2×10 −2 can explain a µ . Taking ν 4 around 400 MeV (800 MeV) and m 5 > m Z ′ , then the Z ′ would decay into 2 ν 4 (ν 4 ν α ) immediately. For the quoted value of the kinetic mixing and the largest neutrino mixing allowed, these heavy fermions would further decay into e + e − and µ + µ − pairs plus missing energy with sub-meter decay lengths. This region of the χ parameter space is constrained only by the BaBar e + e − collider searches for visible [70] and invisible decays [68] of a standard dark photon. Both of these searches would veto the three-body decays of ν 4 , opening up a large region of parameter space (see Ref. [71] for a similar discussion in an inelatic DM model). Resonance searches still constrain the Z ′ BR into e + e − and µ + µ − which are proportional to χ 2 , providing a weak upper bound. In order to shorten the lifetime of ν 4 , we can increase mixing with the tau neutrino in order to avoid constraints from neutrino scattering. A detailed analysis to identify the viable parameter space is required and will be done elsewhere.
Neutrino scattering The presence of a light vector mediator and kinetic mixing can also enhance neutrino scattering cross sections. For a hadronic target Z, the active neutrinos may upscatter electromagnetically into ν h , which subsequently decays into observable particles (ν α Z → (ν h → ν ℓ + β ℓ − β ) Z). Beyond explaining Mini-BooNE, see below, such upscattering signatures can also produce exotic final states in neutrino detectors such as µ + µ − , τ + τ − and multi-meson final states. MiniBooNE low energy excess The above signatures with ℓ ± = e ± have been invoked as an explanation of the excess of electron-like low energy events at MiniBooNE in Ref. [30], where a good fit to energy and angular data is achieved with a similar model containing a single heavy neutrino with m 4 = 140 MeV, m Z ′ = 1 GeV and χ 2 = 5 × 10 −6 . There, the prompt decays of ν 4 were achieved by requiring large mixing with the tau flavour. In a ESSlike limit of our current model, ν 4 would be dominantly produced via upscattering, decaying into ν α e + e − inside the detector. A dedicated analysis to understand the resulting energy and angular distribution is underway.
Dark scalar searches For the scalar portal, the coupling λ ΦH is rather weakly bound by electroweak precision data and the measurement of the Higgs invisible decay at the level of λ ΦH 0.1 [76]. For processes involving λ ΦH , the physical observables are suppressed by mass insertions due to the nature of the Higgs interaction. Nevertheless, if ϕ ′ decays to ν h states, this scalar may also lead to multi-lepton signatures inherited from ν h decays, potentially also in the form of displaced vertices.
In the limiting case of a neutrinophilic model (χ = λ ΦH = 0), the vector and scalar particles present a challenge for detection. Nonetheless, if light, they can be searched for in meson decays [77,78] and at neutrino experiments [79].
Finally, the faster decays of ν h and its self-interactions can help ameliorate tensions with cosmological observations. We do not comment further on this, but note that great effort has been put into accommodating eV scale sterile neutrinos charged under new forces with cosmological observables [22,80] (see also Ref. [81] for an interesting discussion where the Z ′ decay to neutrinos leads to an altered expansions history of the Universe). We note that an eV sterile neutrino with relatively large mixing could be easily accommodated in our ESS framework. The eV neutrino would be mainly in the ν D direction and would have strong hidden gauge interactions.

DARK MATTER
Given the presence of a dark sector, we can ask if the current model can accommodate a DM candidate. This can be achieved introducing new fermions which do not mix with the neutrinos, in order to preserve their stability. A minimal solution would be to introduce a fermionic field ψ L which has U (1) ′ charge 1/2. The different charges of ψ and ν D and N would forbid neutrino mixing. A Majorana mass term ψ T L C † ψ L would emerge after hidden-symmetry breaking leading to a Majorana DM candidate.
Another minimal realisation has the advantage of curing the cancellation of anomalies in the model. Following Ref. [46], we introduce a pair of chiral fermion fields ψ L and ψ R , and charge only the latter under the U (1) ′ symmetry with the same charge as ν D . This choice ensures anomaly cancellation, and allows us to write which after the spontaneous symmetry breaking of U (1) ′ yields a Dirac mass m ψ . In order to avoid ψ R − ν D and ψ L − N mixing, we impose an additional Z 2 symmetry, under which all particles have charge +1, except for ψ L and ψ R , which have charge −1.
If the scalar and vector portal couplings are small in such scenarios, DM interacts mainly with neutrinos. Direct detection bounds are then evaded, since interactions with matter are loop-suppressed. Indirect detection, on the other hand, is more promising as DM annihilation into neutrinos would dominate. For instance, take the mass of ψ to be smaller than the masses of the Z ′ , ϕ ′ and of both heavy neutrinos. In this case, the DM annihilation is directly into light neutrinos via ψψ → ν i ν i . This yields a mono-energetic neutrino line, which can be looked for in large volume neutrino [82,83] or direct detection experiments [21]. On the other hand, if m ψ is larger than the mass of any of our new particles, then the annihilation may be predominantly into such states via ψψ → XX, where X = ϕ ′ , Z ′ or ν h , which subsequently decay to light neutrinos. In this secluded realisation [84], the search strategy for DM can be very different since the neutrino spectrum from such annihilation is continuous [42]. Nevertheless, neutrino-DM interactions are expected to be large and can be searched for in a variety of ways [45,[85][86][87][88].

CONCLUSIONS
We have proposed a new model which invokes the existence of a hidden U (1) ′ symmetry confined to a new dark neutrino sector. The dark sector particles can communicate with the SM via portal couplings, which may be sizeable. The simultaneous presence of neutrino, vector kinetic and scalar mixing in a self-consistent framework allows for a rich phenomenology in present and future experiments, which can be very different from that of each individual portal. In particular, we identified novel signatures such as multi-lepton final states with missing energy, displaced vertices, rare leptonic decays and unique neutrino upscattering processes. We have also argued that existing bounds on heavy neutrinos and dark photons might be significantly weaker as new visible and invisible decay channels appear, opening up previously excluded parameter space. In addition, the model offers a new mechanism for neutrino mass generation and provides a possible connection to dark matter, where the annihilation into neutrinos is the dominant channel.