Neutrino Masses and Mixings Dynamically Generated by a Light Dark Sector

Neutrinos may be the harbingers of new dark sectors, since the renormalizable neutrino portal allows for their interactions with hidden new physics. We propose here to use this fact to connect the generation of neutrino masses to a light dark sector, charged under a new $U(1)_{\cal D}$ dark gauge symmetry. We introduce the minimal number of dark fields to obtain an anomaly free theory with spontaneous breaking of the dark symmetry, and obtain automatically the inverse seesaw Lagrangian. In addition, the so-called $\mu$-term of the inverse seesaw is dynamically generated and technically natural in this framework. As a bonus, the new light dark gauge boson can provide a possible explanation to the MiniBooNE anomaly.


I. INTRODUCTION
One of the most surprising experimental results of the last decades has been the discovery of tiny neutrino masses and relatively large neutrino mixings. Although non-vanishing neutrino masses are a clear indication of physics beyond the Standard Model (SM), the mechanism and the scales responsible for the neutrino mass generation remain a total mystery.
It seems unlikely that the very small neutrino masses are generated by the same Higgs mechanism responsible for the masses of the other SM fermions, since extremely small Yukawa couplings, of the order of 10 −12 , must be invoked. A more 'natural' way to generate neutrino masses involve the addition of new states that, once integrated out, generate the dimension five Weinberg operator This is embodied by the so-called seesaw mechanisms [1][2][3][4]. The smallness of neutrino masses relative to the weak scale implies either that the scale of new physics Λ is very large (making it impossible to experimentally discriminate the different seesaw mechanisms), or that the Wilson coefficient c is extremely small (for instance, coming from loop effects involving singly or doubly charged scalars [5]). A different approach is given by neutrinophilic Two-Higgs-Doublet Models [6,7]. In this framework, a symmetry (U (1) or Z 2 ) compels one of the doublets to couple to all SM fermions but neutrinos, hence being responsible for their masses, while the other Higgs couples to the lepton doublets and right-handed neutrinos. If the second doublet acquires a vacuum expectation value (vev) around the eV scale, this leads to small neutrino masses. These models, however, are either ruled out or severely constrained by electroweak precision data and low energy flavor physics [8,9].
A variation of this idea, in which the symmetry is taken to be a local U (1) and leads to the typical Lagrangian of the inverse seesaw scenario, suffers from an accidental lepton number symmetry that has to be explicitly broken to avoid the presence of a massless Nambu-Goldstone boson in the spectrum [10]. All aforementioned models have one of the two following features: (i) The model is realized at very high scales, or (ii) the model is based on explicit breaking of lepton number or other symmetries that protect neutrino masses (e.g. in TeV scale type II or inverse seesaw models).
Neutrinos, however, are the darkest between the SM particles, in the sense that they can couple through the renormalizable neutrino portal LH operator with generic dark sectors [11]. This fact has been used in connection to thermal Dark Matter with mass in the sub-GeV region (see for instance Refs. [12,13]). In this letter we propose to use such a portal to explicitly connect a new light dark sector with the generation of neutrino masses. In this way, we are able to lower the scale of neutrino mass generation below the electroweak one by resorting to a dynamical gauge symmetry breaking of this new sector. The dark sector is mostly secluded from experimental scrutiny, as it only communicates with the SM by mixing among scalars, among neutrinos and dark fermions, and through kinetic and mass mixing between the gauge bosons. This scheme has several phenomenological consequences at lower energies, and in particular it offers a natural explanation for the long-standing excess of electron-like events reported by the MiniBooNE collaboration [14,15] To avoid any neutrino mass contribution from the Higgs mechanism, we introduce a new dark gauge symmetry U (1) D , under which the SM particles are uncharged, but the new sector is charged. To build a Dirac neutrino mass term we need a SU (2) L singlet righthanded dark neutrino N , and a dark scalar doublet φ, both having the same U (1) D charge +1. The absence of chiral anomalies require a second right-handed neutrino, N , with an opposite U (1) D charge, thus restoring lepton number symmetry. We add to the particle content a dark scalar SU (2) L singlet S 2 , with dark charge +2, whose vev spontaneously breaks lepton number, giving rise to a Majorana mass component for the dark neutrinos. As we will see shortly, this setup leads to an inverse seesaw structure in which the lepton number breaking parameter is promoted to a dynamical quantity. Finally, this scalar content enjoys an accidental global symmetry which is spontaneously broken. To avoid a massless Goldstone boson, an extra dark scalar SU (2) L singlet S 1 , with dark charge +1, is included in the spectrum. Its vev breaks all accidental global symmetries. This field will allow for mixing among all the scalar fields, including the SM Higgs.
The dark scalar S 1 will spontaneously break U (1) D by acquiring a vev, while φ and S 2 will only develop an induced vev after the breaking of the electroweak and dark symmetries. By making a well motivated choice for the hierarchy of the vevs, our model allows a dynamical generation of the light neutrino masses and mixings at very low scale. Our model predicts masses for the dark scalars within the reach of current experiments as well as a light dark vector boson, Z D , that has small kinetic mixing with the photon and mass mixing with the SM Z boson.
The dark particles communicate with the SM ones via mixing: flavor mixing (neutrinos), mass mixing (scalars) and mass mixing and kinetic mixing (Z D ), giving rise to a simple yet rich phenomenology.

A. The Dark Scalar Sector
Let us start discussing the scalar sector of the model. This will motivate the region of parameter space on which we will focus throughout the paper. The most general SU (2) L × U (1) Y × U (1) D invariant scalar potential that can be constructed out of the fields and charges outlined above is (2.1) (In the last sum, the notation ϕ < ϕ is to avoid double counting.) We denote the vevs of the scalar fields as We stress that we are supposing the bare mass terms of H and S 1 to be negative, while we take the corresponding ones for φ and S 2 to be positive. This ensures that, as long as µ = µ = α ≡ 0 (i.e. if there is no mixing among the scalar fields), the latter fields do not develop a vev, while the former do. In turn, this implies that the vevs v φ and ω 2 must be induced by µ, µ , and/or α.
We now observe that µ, µ , and α explicitly break two accidental U (1) global symmetries, making these parameters technically natural 1 . For our purposes, this means that µ, µ and α can be taken small in a natural way, and justifies our working hypothesis v φ , ω 2 v, ω 1 . As we will see later, this hierarchy of vevs will provide a low scale realization of the inverse seesaw mechanism with low scale dynamics associated to it. Explicitly, we obtain approximately being the physical masses of the respective scalars (to be defined below). In order to avoid large mixing between H and φ, we will always make the choice ω 1 v. The scalar spectrum contains, in addition to the SMlike scalar h SM with mass m hSM 125 GeV, three CPeven dark scalars H D , S D and S D , with masses M H D , M S D and M S D , two CP-odd dark scalars A D and a D with masses M A D and M a D , and a charged dark scalar H ± D with mass M H ± D . Explicitly, the masses of the CP-even scalars are 2 4) while the masses of the CP-odd and charged scalars are given by As for the composition of the physical states, since the mixing in the scalar sector is typically small, we can generically define where ϕ physical denotes the physical scalar field that has the largest ϕ admixture. Then, the mixing in the CPeven scalar sector is given by 2 Radiative corrections will naturally contribute to the masses of these scalars. There are potentially several contributions according to Eq. (2.1), the quartic couplings being the most dangerous ones. In order to avoid fine-tuning, we will always demand the masses of the lightest scalars to satisfy M lightest √ λM heavy /8π, where M heavy denotes any of the heavy scalar masses. By the same argument we expect µ, µ and αv to be below M lightest . Our computation ignores the threshold at the Planck scale, which must be stabilized by other means (for instance, supersymmetrizing the theory). 10) We see that our hypothesis v φ , ω 2 ω 1 v prevents any relevant modification to the Higgs-like couplings, and h SM ends up being basically like the SM Higgs boson. Moreover, due to the mixing with the Higgs field, the dark scalars and the longitudinal mode of the Z D will also couple to SM fermions via SM Yukawa couplings. Nevertheless, such couplings to light fermions are quite small as they are suppressed by the hierarchy of vevs. If the spectrum enjoys light degrees of freedom (below the 100 MeV scale), an interesting phenomenology may be associated with this sector. A dedicated study will be pursued in a future work.

B. Neutrino Masses and Mixings
Let us now discuss the generation of neutrino masses and mixings, and how the dynamics of the dark sector outlined so far ensures light neutrinos. The most general Lagrangian in the neutrino sector, compatible with our charge assignment, is where y ν , m, y N and y N are matrices in flavor space. After the two-steps spontaneous breaking SU (2) As already stressed, v φ generates a Dirac mass term, while ω 2 plays the key role to generate a naturally small term y N ω 2 , which can be identified as the tiny mass term of the inverse seesaw µ ISS (the dimensionful parameter of inverse seesaw that breaks lepton number by two units), and we obtain a dynamically generated inverse seesaw neutrino mass matrix. The mass matrix m can always be made diagonal, and in principle take any value, but given the smallness of the Dirac and µ ISS -terms, it is clear that we can generate light neutrino masses even with values of m smaller than that in the usual inverse seesaw scenario. More precisely, the light neutrino mass matrix is given at leading order by in our construction the light neutrino masses are generated effectively as a dimension nine operator (see Fig. 1). Schematically, we start with (2.14) Remembering that the vevs of φ and S 2 are induced by the dynamics of the scalar sector, we can rewrite the previous operator in terms of H and S 1 , the fields whose vev's are present even in the limit {µ, µ , α} → 0. We obtain from which it is clear that, ultimately, neutrinos masses are generated by a dimension 9 operator (see, e.g., Refs. [16] for generation of neutrino masses from higher dimensional effective operators). In addition, we have a further suppression due to the fact that µ and µ can be taken small in a technically natural way. The mixing between active and dark neutrinos can be explicitly written as 16) α = e, µ, τ, D, where ν i and ν α are the neutrinos mass and flavor eigenstates, respectively (we denote by α = D the 6 dark neutrinos flavor states, while U αD is a 9 × 6 matrix). Schematically, we have that the mixing between light and heavy neutrinos is y ν v φ /m. Note that the dark neutrino can be made very light, without introducing too large mixing, even for y ν ∼ O(1) since v φ v.

C. ZD and the Gauge Sector
The new vector boson will, in general, communicate with the SM sector via either mass mixing or kinetic mixing. The relevant part of the dark Lagrangian is (2.17) where m Z D is the mass of Z D , g D is the U (1) D gauge coupling, e is the electromagnetic coupling, g/c W is the Z coupling in the SM, while and parametrize the kinetic and mass mixings, respectively. The electromagnetic and Z currents are denoted by J em µ and J Z µ , while J Dµ denotes the dark current.
In the limits we are considering, the Z and W ± masses are essentially unchanged with respect to the SM values, while the new gauge boson mass reads (2.18) with mass mixing between Z and Z D given by Of course, a non-vanishing mass mixing implies that the Z boson inherits a coupling to the dark current 2.20) While the new coupling allows for the possibility of new invisible Z decays, the large hierarchy v φ v guarantees that the new contributions to the invisible decay width are well inside the experimentally allowed region. The vev hierarchy also protects the model from dangerous K, B and Υ decays with an on-shell Z D in the final state [17,18].
The kinetic mixing parameter is allowed at treelevel by all symmetries of the model. Moreover, it is radiatively generated (see e.g. Ref. [19]) by a loop of the H ± D scalar which magnitude is In what follows, we will take as generated at treelevel, with TREE LOOP to guarantee the radiative stability of the theory. The kinetic mixing will lead to interactions of the Z D to charged fermions, as well as decays if kinematically allowed (see e.g. Ref. [20] for constraints).

III. PHENOMENOLOGICAL CONSEQUENCES
We would like at this point to make some comments about the possible phenomenological consequences of our model. To illustrate the discussion let us consider a benchmark point consistent with our working hypothesis v φ , ω 2 ω 1 v. This point is defined by the input values given in Tab. I, producing the physical observables in Tab. II.
We see that for this point the changes in the masses of the SM gauge bosons as well as the mixings of the Higgs with the new scalars are negligible, so we do not foresee any major problems to pass the constraints imposed to the SM observables by the Tevatron, LEP or the LHC data. Moreover, our model is endowed with all the features needed to explain the excess of electron-like events observed by the MiniBooNE experiment: a new dark vector boson, Z D , that couples to the SM fermions by kinetic mixing and also directly to a dark neutrino, ν D , which mixes with the active ones. As shown in [15], the dark neutrino can be produced via neutrino-nucleus scattering in the MiniBooNE detector and, if m N D > m Z D , subsequently decay as N D → Z D + ν i . The Z D can then be made to decay primarily to e + e − pairs with a rate that results in an excellent fit to MiniBooNE energy spectra and angular distributions.
In general, this model may in principle also give contributions to the muon g − 2 3 , to atomic parity violation, polarized electron scattering, neutrinoless double β decay, rare meson decays as well as to other low energy observables such as the running of the weak mixing angle sin 2 θ W . There might be consequences to neutrino experiments too. It can, for instance, modify neutrino scattering, such as coherent neutrino-nucleus scattering, or impact neutrino oscillations experimental results as this model may give rise to non-standard neutrino interactions in matter. Furthermore, data from accelerator neutrino experiments, such as MINOS, NOνA, T2K, and MINERνA, may be used to probe Z D decays to charged leptons, in particular, if the channel µ + µ − is kinematically allowed. We anticipate new rare Higgs decays, such as h SM → ZZ D , or H ± D → W ± Z D , that depending on 3 Since additional electrically charged/neutral scalar (H ± D , H D , A D ) fields and a light dark gauge boson(Z D ) field are present in our model, they will induce a shift in the leptonic magnetic moments and mediate LFV decays via the interactions as shown in Eq. 2.11 and Eq. 2.17. The contribution to muon magnetic moment from neutral dark Higgs fields (H D , A D ) with flavor-changing couplings is negligible in our framework. The dominant contribution will arise from singly charged scalar (H ± D ) via the interaction term yν L φN . But, the singly charged scalar correction to muon g − 2 is negative and rather destructive to the other contributions. Whereas, the one loop contribution of the dark gauge boson (Z D ) to muon g − 2 is quite promising and a dedicated study will be pursued further on that. It is worth mentioning that there will be another small contribution to muon g − 2 from the W boson exchange via mixing between active and dark neutrinos. m Z D may affect LHC physics. Finally, it may be interesting to examine the apparent anomaly seen in 8 Be decays [21] in the light of this new dark sector.
The investigation of these effects is currently under way but beyond the scope of this letter and shall be presented in a future work.  [14,15]. See Tab. II for the respective physical masses and mixings.

IV. FINAL CONCLUSIONS AND REMARKS
The main purpose of this letter has been to explicitly connect the generation of neutrino masses to the existence of a new light dark sector. Doing so, we are able to lower the scale of neutrino mass generation substantially below the electroweak one by resorting to a dynamical breaking of a new U (1) D dark gauge symmetry under which SM particles are neutral.
Our secluded sector consists of the minimal dark field content able to ensure anomaly cancellation, as well as the spontaneous breaking of the dark gauge symmetry without the appearance of a Nambu-Goldstone boson. It consists of a new scalar doublet, two scalar singlets and a set of six new fermion singlets, all charged under the dark symmetry. A judicious choice of dark charges allows to generate neutrino masses by a dynamical inverse seesaw mechanism, but unlike the usual inverse seesaw scenario, the so-called µ ISS -term is here dynamically generated, and can be small in a technically natural way. Interestingly, neutrino masses effectively emerge only at the level of dimension 9 operators, and we can have a new light dark gauge boson in the spectrum.
The dark sector is mostly secluded from experimental scrutiny, as it only communicates with the SM by mixing: the SM Higgs mixing with dark scalars, neutrinos mixing with dark fermions, and through kinetic and mass mixing TABLE II. Physical masses and mixings for the benchmark point of our model that can provide an explanation of the low energy MiniBooNE excess [14,15]. The light-heavy neutrino mixing is schematically denoted by |UαN | 2 , and mN D denotes the order of magnitude of the diagonal entries of the dark neutrino mass matrix.
with the dark gauge boson. The low scale phenomenology of the model is simple yet rich. It is possible that our model gives sizable contributions to several experimental observables such as the value of the muon g − 2, the Majorana mass in neutrinoless double β decay or influence atomic parity violation, polarized electron scattering, or rare meson decays, among others. Moreover, the mechanism we propose in this letter could provide an novel explanation to the Mini-BooNE low energy excess of electron-like events [15].
As a final remark, let us stress that we presented here only the low scale realization of the model, imposed by the hierarchy of vevs we have selected. Nevertheless, we could have chosen a different one, for instance, ω 1 v. In that case we would have a high scale realization of the model, with unique phenomenological consequences at the LHC, for instance displaced vertex or prompt multilepton signatures.