Neutrino and $Z'$ phenomenology in an anomaly-free $\mathbf{U}(1)$ extension: role of higher-dimensional operators

We consider an anomaly-free $\mathrm{U}(1)$ extension of the Standard Model with three right-handed neutrinos (RHNs) and two complex scalars, wherein the charge assignments preclude all tree-level mass terms for the neutrinos. Considering this setup, in turn, to be only a low-energy effective theory, we introduce higher-dimensional terms {\em a la} Froggatt-Nielsen to naturally generate tiny neutrino masses. One of the RHNs turns out to be very light, thereby constituting the main decay mode for the $Z'$ and hence relaxing the LHC dilepton resonance search constraints. This very RHN has a lifetime comparable to or bigger than the age of the Universe, and, hence, could account for a non-negligible fraction of the dark matter.


Introduction
Of all the lacunae besetting the Standard Model (SM), the existence of Dark Matter (DM) and the flavour problem are rather vexing ones. While the masses of the charged fermions as well as the mixing amongst the quarks can be explained by postulating a certain set of Yukawa couplings, the large hierarchy between these is a rather disquieting feature. Over the last few decades, several disparate sets of theories have been proposed including (but not limited to) (i) additional symmetries, discrete or continuous, gauged or global [1][2][3][4][5][6][7][8][9][10][11], (ii) quark compositeness, whether in terms of further constituents, bound together by some unspecified force [12][13][14][15] or, in the more modern parlance, in terms of a higher-dimensional theory, often with a nontrivial gravitational background [16][17][18][19][20]. Very often, though, such efforts are faced with undesirable phenomenological consequences (unsuppressed flavourchanging neutral currents being one such), and these issues can be cured only through the introduction of further complications. Even more damagingly, no corroborating evidence has been found in terms of additional particles (that many such scenarios posit) or interactions.
The situation has been exacerbated in recent years by the observation of neutrino oscillations [21][22][23][24][25] and these seemingly call out for nonzero neutrino masses. Indeed, the consequent mixing angles in the neutrino sector are quite well-determined and so is one difference in the squares of masses [26,27]. For the other independent difference, only the magnitude (and not the sign) is known and also unknown are the nontrivial phases that are possible in the mixing matrix. And while the oscillation data is only sensitive to the difference in mass-squareds and not the absolute mass scale, the latter is very-well constrained to m(ν i ) < (0.340−0.715) eV [26] -where ν i are (cosmologically) stable light neutrinos -from a host of cosmological data. Interestingly, direct bounds from terrestrial experiments (such as decays) are significantly weaker [28].
Of course, neutrino masses (absent within what is known as the SM) can be trivially obtained by introducing right-handed neutrino (RHN) fields ν jR and ascribing additional Yukawa terms. Tuning the said couplings so as to obtain the requisite masses and mixings is a seemingly trivial task, once a further hierarchy in the Yukawa couplings (necessitated by the smallness of the neutrino masses) is accepted. The pitch is further queered, though, by the fact that with the RHNs being gauge singlets, terms such (ν jR ) c ν kR are gauge invariant, and being unprotected by any symmetry 1 , can be arbitrarily large. Beset with such large Majorana masses, the ν jR can be integrated out from the low-energy theory, leaving the SM neutrinos with tiny masses. Indeed, this very observation led to a cure in the form of the seesaw mechanism, wherein a large (dynamical) scale is set for the Majorana mass matrix m M , with and the usual Dirac mass matrix (m D ) for the neutrinos being unsuppressed, so that on integrating out the heavy fields, the light fields are left with an effective mass matrix m eff ∼ m T D m −1 M m D which, on diagonalization, should yield the observed masses and mixings.
With m D being proportional to the vacuum expectation value of the SM Higgs H, the aforementioned structure could have been divined in an effective theory. Written in a gauge-invariant form, the Weinberg operator [29] reads c ij L c i L j HH/Λ where L i are the lefthanded lepton doublets and Λ is the cutoff scale (perhaps allied to the largest eigenvalue of m M above). The dimensionless constants c ij constitute a symmetric matrix and can be thought of as parametrizing the structure of m T D m −1 M m D . Once again, choosing c ij /Λ to be small enough and ascribing the necessary structure, the correct set of masses and mixings can be obtained.
All of the aforementioned mechanisms to generate the light neutrino masses "suffer" from the requirement of either postulating very tiny couplings or a very large scale (e.g., m M ∼ 10 11 GeV for the so-called "type-I" seesaw mechanism) rendering such theories essentially untestable by current experiments. It would, thus, be very attractive to have a theory for neutrino masses with a characteristic scale O(10 TeV) or lower so that it is testable at the LHC, the B-factories etc.. Examples are scenarios [30,31] of TeV scale RHNs with a significant mixing with the SM νs achieved through the realization of low scale seesaw through non-trivial flavor structure. Similarly, models with radiative neutrino mass generation and/or inverse seesaw [32] also exist; with relatively light RHNs, these can be probed at colliders. Looking for well-motivated scenarios that incorporate experimentally testable RHNs in a more broader scheme, is the goal we set for this work.
Before we start this in right earnest, it is worthwhile to remind ourself of a particularly elegant proposal addressing the fermion mass hierarchy. As Froggatt and Nielsen (FN) [33] pointed out, ascribing the Higgs and quark fields with some extra charges (corresponding to a discrete or a continuous symmetry) would, in general, render the usual Yukawa terms untenable. Instead, higher dimension terms could be written by inserting an appropriate number of a "flavon" scalar field F. The choice of the charges would dictate the powers of the ratio F /Λ (Λ being the cutoff scale) in the effective mass terms and hence their scales. Apparently, then, this simple ruse can lead to correct masses and mixings without the need for imposing a large hierarchy in the Yukawa couplings [34]; and the non-renormalizable nature of the theory could be explained as being the result of integrating out unspecified fields, the nature of which would depend on the particular ultraviolet completion of the FN-scenario. Unfortunately, though, the simplest such models turn out to be phenomenologically inconsistent, failing to satisfy the constraints from rare decays while still explaining the masses and the mixings.
While we would not discuss charged fermion masses in this paper, it is still instructive to examine the FN mechanism and, in particular, where the flavor charges correspond to a U (1) symmetry. An extra gauged U(1) can, of course, appear in many a scenario, ranging from flavor models to theories of compositeness to GUTs . Naturally, the phenomenological consequences are very well studied [56][57][58][59][60][61][62][63][64][65][66][67] and strong constraints (in the mass-gauge coupling plane) emerge from a variety of measurements, ranging from rare decays, anomalous magnetic moments of the electron or muon, electroweak precision tests (performed at the Z-peak) to direct observation at the LHC. The relative strengths of the constraints are determined by the U(1) charge assignments for the light SM fermions. The latter, of course, are not entirely arbitrary as the U(1) extension needs to be anomaly-free. In particular, some of the strongest constraints emanate from the lack of unexplained, yet discernible, peaks in the invariant mass spectra for dijet or dilepton production at the LHC.
In this paper, we pursue a modest goal. Starting with an anomaly-free U(1) extension of the SM (augmented by the mandatory three RHN fields), we employ a Froggatt-Nielsenlike mechanism, but restricted strictly to the neutrino sector. We find that (a) the neutrino masses and mixings can be explained with very moderate choices for the Yukawa couplings and a U(1) scale of a few TeVs; (b) simultaneously, the dilepton branching fraction of the Z is suppressed so that even with very natural choices of parameters, a Z as light as 3 TeV is perfectly consistent with the LHC results and, yet, (c) novel signatures are predicted at the LHC.
An additional bonus is the natural emergence of a viable Dark Matter candidate, thereby addressing the second (and, perhaps, even more pressing) lacuna of the SM that we had alluded to. In fact, as many as two ultralight particles (a pseudoscalar and a RHN) appear in the spectrum, with their masses uplifted only by higher-dimensional operators.
While not strictly stable, the lighter of these has a lifetime comparable to or greater than the age of the universe. Being charged under the extra U(1) (although neutral under the SM gauge group), their interactions are large enough to be interesting in the context of both cosmology and direct detection. The rest of the paper is organized as follows. We discuss the anomaly-free U(1) z extension model in Section 2 including gauge, scalar and fermionic sectors of the model. We devote the phenomenology of the neutrino sector and Z boson in Sections 3 and 4 respectively wherein we discuss various exclusion limits. In Section 5, we explore the possibility whether the new ultralight particles present in our model can act as a suitable dark matter candidate. Finally, we summarize and conclude in Section 6.

The U(1) z extension
The gauge sector of the SM (i.e. SU(3) c × SU(2) L × U(1) Y ) is extended by a new U(1) z gauge symmetry with the associated gauge coupling g z . Presence of multiple U(1)s in a gauge theory can, in general, lead to kinetic mixing. However, it is always possible to rotate away this kinetic mixing at a given scale (due to running of the couplings, it can be regenerated at other scales). In this paper, we are only interested in the effective TeV-scale phenomenology and therefore, we make the simplifying assumption that the kinetic mixing between U(1) Y and U(1) z is removed by a suitable field rotation at the TeV-scale. We further assume that the SM fields are charged under the new U(1) z and the corresponding quantum number of a field F being denoted by z F . With the SM fields too allowed to have nonzero z F , anomaly cancellation is a concern. Postponing this concern until later, we begin by considering the bosonic sector.

Scalars and symmetry breaking
At a scale much higher than the electroweak symmetry breaking (EWSB) scale, the U(1) z is broken by one or more SM singlets χ A carrying charges z χ A . While a single χ suffices for the requisite symmetry breaking, we generalize the situation for phenomenological reasons which will be clear later. The Lagrangian for the scalar sector is given by where H denotes the SM Higgs doublet and our assumption that the charges z χ A are such that trilinear terms in the potential are not admissible would be vindicated later. The covariant derivatives, for a generic field F is, of course, given by with X µ being the new gauge boson. The second, third and the fourth terms on the r.h.s. of the first of the above equations correspond to the SU(3) c , SU(2) L and U(1) Y gauge groups of the SM with gauge couplings g s , g w and g y respectively.
The symmetry is broken, in two steps, by the vacuum expectation values (vevs) of χ A and H fields.
We assume that, in case multiple χ A are invoked, the corresponding vevs x A are of the same order, i.e., no hierarchy is introduced between them. Furthermore, for the sake of simplicity, we do not admit spontaneous CP violation, or in other words, nontrivial phases between the vevs. While the expression for W -boson mass M W remains unchanged, the (mass) 2 matrix for the neutral gauge bosons is now modified to Clearly, M 2 is a rank-2 matrix, and can be diagonalized by an orthogonal matrix O defined through The Weinberg angle w remains unaltered, namely w = tan −1 (g y /g w ) whereas for e = g w sin w, the Z ↔ Z mixing angle t is given by The heavy neutral gauge boson masses are given by The shift in M Z imposes a constraint on the parameter space of the model, which, as far as this sector is concerned, could be considered of being three-dimensional, namely defined by g z , z H and the combination A z 2 χ A x 2 A . Note, however, that the above treelevel expression cannot be immediately compared with the experimentally measured M Z as quantum corrections need to be included. We return to this point later.
Another example of such changes would be that wrought by the scalar sector. The very inclusion of a single χ A field results, after the breaking of the U(1) z and in the unitary gauge, in an additional scalar field. Although a SM singlet, this can mix with the SM Higgs field (owing to terms such as H † Hχ † χ) resulting in two physical scalars h 1,2 . It thus needs to be ensured that at least one of the two eigenstates has a mass of 125 GeV and couplings (both gauge and Yukawa) not significantly different from the SM Higgs. As it would turn out, this holds almost trivially for the parameter space that we are interested in. It should also be apparent that once one of these (say h 1 ) is forced to be very SM-like, the other (h 2 ), being singlet-dominated, would have very small production cross sections at the LHC. The modes of interest would be qq → Z h 2 (analogous to the Bjorken process) and gg → h 1 h 2 with (h 2 -sstrahlung). Understandably, such an h 2 would have expected detection thus far.
The introduction of each additional χ A results in the physical spectrum being enhanced by a pair of spin-0 particles, one scalar and one pseudoscalar. Particles in each sector could mix amongst themselves with inter-sector mixing disallowed as long as additional CP -violation (explicit or spontaneous) is not introduced in the Higgs sector. It should be noted that this introduces a new class of modes, namely the Z (on-shell or off-shell) going to a scalar-pseudoscalar pair.

Example with two χ fields
As an example, consider the special case of there being two such fields χ 1 and χ 2 . This would prove to be of particular interest in the context of neutrino mass generation. If the corresponding U(1) z charges z χ 1 and z χ 2 are not integral multiples of each other 2 , the most general form of the scalar potential invariant under U(1) z is given by Here, we have deliberately suppressed terms containing the SM Higgs field. This simplifying approximation, apart from being very good at energies far above the electroweak scale, serves to highlight certain salient features. As is immediately apparent, V (χ 1 , χ 2 ) is invariant under a global U(1) × U(1) with each factor associated with one of χ 1,2 . The most general symmetry breaking would, thus, result in two Goldstone fields G 1,2 . One linear combination of the two U(1)s is gauged, viz. U(1) z ∼ z χ 1 [U(1)] 1 + z χ 2 [U(1)] 2 , and the corresponding combination of G 1,2 appears as the longitudinal component of B µ . The orthogonal combination appears in the spectrum as a massless pseudoscalar A. While it may seem that the presence of a Goldstone in the theory would render it phenomenologically nonviable, this is not necessarily so, as we argue below.
A true Goldstone would translate to a long-range force. And, even if it acquired a small mass through an explicit symmetry-breaking term, such a particle would still make its presence felt in a variety of interactions. Some of the strongest bounds emanate from low-energy processes including, but not limited to, astrophysical ones. Indeed, Majoron models (i.e., ones, analogously to us, wherein a Goldstone arises in spontaneously breaking a lepton number symmetry in the quest to achieve Majorana masses [68] have been wellstudied in this context. As it turns out, though, in the present case, the Goldstone would have very suppressed couplings with the SM fermions (thanks to the quantum number assignments) and, consequently, such bounds are expected to be satisfied easily. Indeed, the only SM particle that it might have unsuppressed couplings with is the Higgs, thanks to possible terms such as Post EWSB, these immediately give rise to a trilinear HAA coupling, with a strength determined by the λ Hχ 1,2 . It is interesting to note that the data on the 125 GeV scalar still allows for a non-significant invisible decay of the particle [69] and this can be used to implement an upper bound on these quartic couplings. It should be realised, though, that the Goldstone would, in general, be lifted by quantum corrections, rendering it a pseudo-Nambu-Goldstone Boson (pNGB). For example, consider an effective theory exemplified by the inclusion of higher-dimensional terms in the Lagrangian parametrizing unknown effects emanating from physics at still higher energies. While Eq. (2.8) represents the most general gauge-invariant potential consistent with renormalizability, once nonrenormalizable terms are allowed, more terms can be present. In the next section, we would argue for z χ 1 = −3/4, z χ 2 = −4, and for such a case the lowest-dimensional term that breaks the global [U(1)] 2 down to U(1) z is χ 16 1 χ * 3 2 . Such a large engineering dimension of the operator would, typically, generate only a very small mass for the pNGB viz. O(x 16 i /Λ 15 ) where Λ is the cutoff scale. For x i /Λ < ∼ 0.1, a very reasonable restriction, this would leave the pNGB in a milli-eV range, reminiscent of axionic dark matter models.
It needs to ascertained, though, whether χ 1,2 are allowed to have gauge-invariant Yukawa terms involving any new fermions in the theory (such as RHN fields) and whether these explicitly break [U(1)] 2 down to U(1) z . If such be the case, the pNGB would be lifted courtesy loop corrections 3 .
Were it desirable to substantially raise the Goldstone, it could be trivially done through the introduction of a third singlet scalar χ 3 , which, in principle, could actually increase the global symmetry to [U(1)] 3 . On the other hand, this may allow for renomalizable terms breaking down the global symmetry to just a single U(1) to be identified with the gauged U(1) z . For example, working with the previously assigned quantum numbers for χ 1,2 , if one introduces a third scalar χ 3 with a charge 13/4, then a term such as (µ 123 χ * 1 χ 2 χ 3 + H.c.) would break the symmetry softly. On the other hand, if z χ 3 = 5/2, then a hard breaking is achieved through (λ 1123 χ 2 1 χ 2 χ * 3 + H.c.). It is easy to see most such augmentation of the scalar sector does not materially alter low energy phenomenology except, perhaps, to ameliorate some issues with the evolution history of the early universe. As far as LHC signals go, the most drastic changes would be in the decays of the Z wrought by the proliferation of states (three and two SU(2)singlet scalars and pseudoscalars). Since these effects are easily computed and are not very germane to the issues that we are primarily interested in, we will not discuss such a three-singlet scenario any further.
To start with let us work under the assumption that the χ-sector is essentially decoupled from the SM Higgs sector. This is not too drastic an approximation at energy scales much higher than the electroweak scale (except as far as the decays of the χ i into the SM Higgs is concerned). The potential, then, is described by that in Eq. (2.8). Denoting the χ fields, post symmetry breaking, by where ξ 1,2 , ρ 1,2 are real fields, the massless pseudoscalar is given by with the orthogonal combination being absorbed to reappear as the longitudinal mode 4 of the Z . The mass-squared matrix for the two scalars ξ 1,2 reads leading to mass eigenstatesξ 1,2 defined by with the corresponding masses being

Fermionic sector and anomalies
Since one of our primary goals is to explain neutrino masses and mixings, we must include extra neutral fermions and at least two of them. This is the only addition we propose in this sector, and as we would shortly see, invoking three such right-handed fields is not only enough to ensure the cancellation of all possible anomalies 5 , but also leads to very interesting phenomenological consequences. For the sake of simplicity, we consider the U(1) z charges to be family-blind, as far as the SM fermions are concerned, denoting these to be z Q (quark doublets), z L (lepton doublets), z u , z d (the right-handed up-like and down-like quarks respectively) and z e for the right-handed charged leptons. Similarly, three RHN fields N i are assigned charges z i , not necessarily equal. Before we consider the fermion masses and, thereby, relate these to z H and z χ A , let us first discuss the anomalies defined as where T a are the symmetry generators and the traces are over left-and right-handed fermions. The SM gauge anomalies, of course, remain unaltered and the only nontrivial quantities are those pertaining to U(1) z and are listed below 4 With the SM Higgs not yet acquiring a nonzero vacuum expectation value, there is no Z-Z mixing at this stage. 5 While, gauge anomalies can also be canceled using the Green-Schwarz mechanism-we refer the reader to Ref. [70] for a phenomenological discussion of such constructions-we eschew this in favour of a more canonical approach. Anomaly It is easy to see that, using the first two conditions, the third simplifies to 2z Q +z u +z e = 0. Similarly, the first three, together, imply that the fourth one is satisfied identically. And, finally, the fifth one simplifies It is also easy to see that the mixed gauge-gravity anomaly (R 2 U(1) z ) does not present an independent constraint.

U(1) z charge assignment
The existence of mass terms for the charged fermions demands that Note that only one of these equations is independent once anomaly cancellations have been imposed (in fact, just the [SU(3) c ] 2 U(1) z and [SU(2)] 2 U(1) z are enough). The U(1) z charges of the SM fields can, then, be expressed in terms of just two parameters, say z u and z Q . Note, however, that, for any U(1) theory, one combination of charges can always be taken to be unity, without any loss of generality. In the present case, we shall choose z u − 4z Q = 1 and consider z Q to be the remaining free parameter. The consequent charge assignments have been displayed in Table.1. The charges z i for the N iR fields, thus, need to satisfy and the solution space is a two-dimensional one. Restricting ourselves to rational values, the simplest assignment would be z i = 1, a choice that has been explored in a different context [71]. This, though, is unsuitable as far as neutrino mass generation is concerned. Consequently, we adopt the next simplest choice, namely z 1,2 = 4, z 3 = −5.

Neutrino masses
While the Yukawa (and, hence, the mass) terms for the charged fermions proceed as within the SM, viz., where H = iσ 2 H * , dimension-4 gauge invariant Yukawa (or even bare mass) terms are not possible for the neutrinos. The situation changes if the theory is treated not as a fundamental one, but only as the low-energy limit of some more fundamental theory operative where the couplings y iα ,ỹ i , w ij , s αβ , s α3 and s 33 are dimensionless and the exponents satisfy It should be realized that only integer solutions for the exponents are permissible as nonintegral values would imply nonlocal operators. Negative values for the exponents are to be interpreted as positive powers of χ * 1 (χ * 2 ) as the case may be. Before discussing the ramifications of L νmass , it is instructive to remind ourselves of the possible origin of the same. As can be readily recognized, these can arise from a UV-complete theory once a slew of fields (especially fermionic ones) are integrated out. Clearly, these fields must have masses larger than Λ and carry nonzero U(1) charges. It might be argued, then, that the requirement of the effective theory being anomaly-free is a superfluous one, for the anomaly(ies), being a child of the UV regularization, could, in principle, be canceled by the heavy fermions Ψ i . However, note that the Ψ i themselves should be vector-like, as else their masses can only arise from spontaneous breaking of the U(1) symmetry and, thus, should be below Λ. And since vector-like fermions do not contribute to gauge anomalies, the effective theory better be anomaly-free.
A further issue pertains to the relative strengths of tree-order and loop-level contributions in the generalization of the FN-mechanism that our theory really represents. Consider a typical term in L νmass which has n powers of, say, χ 1 . Letting the χ 1 lines go into the vacuum (courtesy spontaneous symmetry breaking) gives us a factor of (x 1 /Λ) n . On the other hand, closing a χ 1 loop would, typically, give us a factor of O(Λ 2 /16π 2 x 2 1 ), as the loop momentum would need to be cut off at the scale Λ. Thus, the exclusion of loops is well motivated for Λ > x j > ∼ Λ/(4π), inequalities that we would satisfy in further calculations.
With each of the terms in L Wein. violating lepton-number, it is tempting to characterize the corresponding mass terms, realised on breaking the U(1) z symmetry, as Majorana masses. An argument against this would be the fact that, in the canonical sense, a Majorana particle may not have any nonzero additive quantum numbers, whereas each of ν iL and N iR certainly do. Rather, these terms should be thought of as the generalization of the Weinberg-operator [29] that is allowed as a dimension-5 correction to the SM. Indeed, on the breaking of the U(1) z symmetry, the corresponding charge is no longer a valid quantum number in the ensuing theory, and the mass term generated thereupon can indeed be thought of as a Majorana mass.
As a very specific case, let us consider the assignment which leads to rather interesting phenomenology 7 . With this choice, the masses for N 1,2 sub-sector are relatively unsuppressed, while terms connecting N 3R are highly suppressed. Indeed, retaining just the least suppressed terms in each sector would lead to The terms corresponding to w ij and s 33 are too small to be of any consequence, and, formally, could be dropped altogether if we restrict ourselves to operators of mass dimension 7 It should be realised that this choice is not a special one and qualitatively similar results would be obtained for many other choices. 5) and this is the form that we would be working with henceforth.

Identifying the mass eigenstates
The neutrino mass matrix, as given in Eq. (3.5) can be represented, in the (ν j , N 3 , N 1 , N 2 ) basis, by Denoting ξ ≡ x/Λ, where x is either of x 1,2 (we assume that there is no large hierarchy between the x i ), the matrices above have structures Note that the N 1 -N 2 sub-sector of the mass matrix can be chosen to be diagonal without any loss of generality, with a 1 and a 2 as the coefficients. While two eigenvalues of M N are large, viz. O(x 2 /Λ), these heavy neutrinos still tend to be lighter than the Z , owing to the smallness of ξ (in comparison to g X ). The third eigenvalue of M N is much smaller, namely only λ 3 ∼ O(ξ 6 x 2 /Λ). This state has only a small mixing with the heavier ones, with the mixing angles being O(ξ 4 ).
More importantly, it might seem that a straightforward application of the seesaw mechanism may not be possible, given that one of the eigenvalues of M N is smaller than some of the Dirac masses. However, note that the elements D i1 are smaller than the corresponding fulcrum λ 3 by at least a factor of vξ/x. Consequently, the seesaw mechanism goes through trivially, and even the first-order estimate is rather accurate. This is corroborated by a full numerical calculation of the eigensystem of the full 6 × 6 mass matrix.
On block-diagonalization, we have the effective mass matrix for the light-sector to be given by Explicitly, Several points demand attention: • As argued above, dropping the higher order terms in in M 3×3 is an excellent approximation. Similarly, the mixing between the ν i and N α is quite small and the eigenstates of M 3×3 are predominantly doublets.
• If the Yukawa couplings are real (i.e., if there is no CP violation in this sector), then the real symmetric matrix M 3×3 is given in terms of 6 parameters. These can be uniquely determined in terms of the three light neutrino masses and the three mixing angles.
• A priori, the matrix D is defined by nine parameters (assumed to be real), and further parameters appear in M N . However, only six independent combinations may enter M 3×3 , with the rest serving only to define the heavy sector and the tiny mixing between the heavy and light sectors.
• Finally, while exploring the parameter space, it is important to ensure that (M 3×3 ) ee is bound by the non-observation of neutrinoless double beta decay [72].

Neutrino Results
Since the heavy-light mixing is tiny, low-energy experiments are well-described in terms of the light neutrinos alone. To this end, we diagonalize the light part of the mass matrix through where the PMNS matrix U is given by with R jk denoting a rotation in the jk plane through an angle θ jk . In other words, where c ij ≡ cos θ ij and s ij ≡ sin θ ij .  Table 2. Bounds on neutrino mass and mixing parameters from neutrino oscillation experiments, for both normal hierarchy (NH) and inverted hierarchy (IH) of the neutrino mass spectrum. The error bars shown correspond to 1σ [27].
The different Yukawa parameters are varied such that the mass squared differences of neutrino mass eigenstates namely m i , i = 1, 2, 3 defined as, along with three mixing angles θ 12 , θ 13 , θ 23 conform to the experimental results as listed in Table. 2. Another important constraint which is to be satisfied is cosmological one on the sum of the masses of the light stable neutrinos, namely, [73]: with 95% confidence level (CL). A more stringent limit can be obtained if we take into account Baryonic Acoustic Oscillations [74,75]. But we have stuck to the limit above. Clearly, we have more parameters in the theory than there are data in the neutrino sector. Consequently, some parameters are indeterminable, and must be fixed by hand. To ease the task of identifying the crucial dependence, we choose to make some simplifying assumptions: • While we had already assumed that there was no large hierarchy between the vacuum expectation values for the two new scalars, we eliminate any choice and consider only x 1 = x 2 = 10 TeV.
• The two heavy RHNs, predominantly N 1,2 , should have a mass O(ξx) and these are held at 1.2 TeV and 1.25 TeV respectively. The splitting between them is not germane to the discussion at hand, and has been incorporated just to ensure that the numerical results are not affected by degeneracy.
• The third RHN (N 3 -like) is held at 6 keV. This can be achieved, for example, if s α3 ≈ 0.05. in Eq. (3.4). As it turns out, certain results are quite dependent on this (although, not the light neutrino phenomenology) and we shall return to this in a later section. It should be realized that the above choices are not special in any way and do not leave qualitative impact on the determination of the rest of the parameter space. These are to be fixed by an analysis of the neutrino oscillation results. Clearly, this last bit would be dependent on the two different neutrino mass hierarchies that are experimentally viable, and we consider them in turn. In doing this, it needs to be borne in mind that, for a very large part of the parameter space, one of the three SM-like neutrinos is distinctly lighter than the others. This is but a consequence of the fact that, were N 3 to be decoupled, the light mass matrix M 3×3 (resulting from the seesaw mechanism) would be a rank-2 one.

Normal Hierarchy
Terming the lightest mass m 1 ∼ 0, the two other mass eigenvalues would be m 2 ∼ √ ∆ 21 and m 3 ∼ √ ∆ 32 + ∆ 21 , thereby satisfying the hierarchy m 1 < m 2 m 3 . As is well known, the requirement of θ 23 ∼ 45 • imposes strong constraints on the neutrino mass matrix, and is often sought to be explained by family symmetries. In the present context, this is to be ensured by judiciously choosing the couplings y ij guided by Eq. (3.10). While many different solutions are possible, given that ξ is small, if we want to eschew very large hierarchies in the couplings and/or large cancellations, we should look for the possibility that y 21 − y 22 ≈ y 31 − y 32 ∼ 0.01. The first approximate equality ensures that the leading contributions are of the same order, while the second one ensures that cancellations are not extreme (since the couplings themselves would turn out to be of a similar size). The correct value for the heaviest neutrino mass m 3 is obtained for a somewhat larger value of y 33 , namely y 33 ∼ O(0.1). As Fig. 1 shows, it is possible to satisfy all the neutrino constraints in our framework with these Yukawas, without resorting to very small values as is often the case for general seesaw models. Indeed, relatively larger values of the Yukawas y 21 ∼ 0.08, y 22 ∼ 0.03 as shown in Fig. 1 play a pivotal role to have a significant mixing in the 1 − 2 sector. It should be realized that Fig. 1 does not reflect the entire viable parameter space. The very structure of Eq. (3.9), for example, stipulates that the interchange (y 11 , y 21 ) ↔ (y 12 , y 22 ) would result in identical masses and mixings. Rather, the parameter space displayed in Fig. 1 should be considered a representative set of solutions.

Inverted Hierarchy
Similar to the normal hierarchy, we have taken a simplified scenario where m 3 is now taken to be zero. Other two mass eigenvalues in this case then become m 1 ∼ √ ∆ 23 − ∆ 21 and m 2 ∼ √ ∆ 23 . This satisfies the hierarchy m 3 m 1 ≈ m 2 . Similar in the previous case, some initial assumption on Yukawa couplings are taken such as y 11 = y 21 + 0.01 and y 12 = y 22 + 0.01, such that we have similar contribution from (y 21 − y 22 ) and (y 11 − y 12 ). This is required to satisfy the mass of m 1 to be non-zero along with the significant mixing with θ 12 ∼ 33 o , in the 1 − 2 sector. There are two more distinct regions of Yukawas with magnitude of similar order to those shown in the plots where all the neutrino constraints are satisfied. In this case also, the allowed Yukawas values satisfying the neutrino results are significant enough, as presented in the Fig. 2. In the allowed parameter space, the difference y 31 − y 32 is relatively larger than the difference y 21 − y 22 , as depicted in the Fig. 2, helps to have a negligibly small m 3 term along with significant mixing in the form of θ 23 ∼ 45 o .

Z phenomenology
Until now, we have investigated the parameter space of the model only in the context of the light neutrino masses and mixings. As is evident, these observables are sensitive primarily to the Yukawa couplings (the Wilson coefficients in Eq. (3.5)). While there is a dependence on x i and Λ, these appear in a trivial fashion, and as far as the limited number of observables available at low energies are concerned, these dependencies can be entirely subsumed in the Wilson coefficients 8 . And, finally, this sector carries virtually no imprint of either the gauge coupling g z , nor the parameters in the Higgs potential.
The aforementioned parameters are best investigated at colliders, either through direct production or by effecting precision studies. Seven additional physical states now appear: a massive neutral gauge boson Z , two massive scalars H 1 and H 2 , a relatively light pseudoscalar A, two massive (predominantly right-handed) neutrinos N 1,2 and, finally, a light neutrino N 3 (again, predominantly right-handed). As for the parameters in the gauge sector, apart from g z and x i , we also have the hitherto unfixed charge z Q (see Table 1), in terms of which the U(1) z charges of all the SM fields were specified, courtesy the requirements of anomaly cancellations. That z Q is still free is but a reflection of the fact that U(1) charges are, intrinsically, not quantized unless the symmetry had descended from a bigger group.
With the Z having a substantial coupling to all the SM fields 9 , production of the Z at, say, the LHC would be expected to constitute a sensitive probe of the scenario. Interesting signatures of heavy neutrinos of our model can also be searched for at future lepton colliders [76]. Before delving into the details of the Z phenomenology, it is amusing to note that choosing z Q , hitherto a free parameter, appropriately would minimize the production cross-section without affecting the neutrino phenomenology. For example, to the leading order, Here, X symbolizes the rest of the hadronic byproducts and the flux F u is given by with √ s being the total center-of-mass energy available at the LHC, and f j (x, Q 2 ) the density of the j th parton for a given momentum fraction x and computed at the scale Q. An analogous expression holds for F d . While the ratio of the two fluxes (those for  the other quark flavours, being subdominant, result in only a very minor correction) is a function of M 2 Z , it is intriguing to note that, for most of the region of interest, σ(pp → Z + X) is minimized for z Q ∼ −1/4. It is interesting to note that a choice of z Q = −1/4 is commensurate (apropos U(1) z charge quantization) with our assignments for the z χ i . Consequently, we use this value of z Q for much of the numerical calculations below. However, this choice is not a very crucial one and somewhat removed values of z Q might also be chosen as illustrated by an alternative choice for some of the figures.

Branching ratios
At a collider, of all the new states, the production of the Z is the easiest. Consequently, we begin by considering its branching fractions. Owing to its large mass, the decay is dominated by the two-body modes and these can be calculated trivially. We analytically compute various partial widths of two-body decay modes of the Z and the expressions thereof are given in Appendix 7. Our numerical results are cross-checked against Mad-Graph [77].
As a particular benchmark point in the parameter space, we consider Given the large U(1) z charge of the N 3 and its small mass, it is understandable that the Z →N 3 N 3 mode overwhelmingly dominates. While, individually, the Z →N 1,2 N 1,2 are expected to be suppressed (compared to theN 3 N 3 one) by only a factor of 16/25, these also suffer an additional kinematic suppression, especially at lower values of M Z . For high Z masses, together, these would slightly overcome the former mode. Of the decays into exclusively SM decay modes, of particular interest (since the corresponding SM backgrounds are not too large) are the ones to W + W − and ZH SM . Equal on account of the Goldstone equivalence theorem, these tend to be small as their amplitudes are proportional to the Z ↔ Z mixing. In particular, for z Q = −1/3, the coupling of the SM Higgs H to the Z vanishes identically (at least at the tree level), and so does the Z-Z mixing and, hence, these modes vanish too 10 . Less suppressed are the modes Z → (H 1 + H 2 )A.
That the decays into exclusively SM modes tend to be subdominant for |z Q | < ∼ 1/3 is but a reflection of the charges, which also accounts for the ratios of the said partial widths. This suppression, though, does not hold for larger values of |z Q |, as is reflected by Fig. 3c. Indeed, for a sufficiently large |z Q |, it is the dilepton or dijet modes that dominate. It is interesting to note that the value of z Q that minimizes Z production is not the same as when these decay modes minimize. While we could have chosen a z Q such that, say, the dilepton signal at the LHC is minimized (rendering the model relatively free from constraints), we choose not to do so.
It is also worthwhile to consider the decay modes of N 1,2 . The exact details would, of course, depend on the specific pattern of the Yukawa couplings. The decay to W ± and ± is higher than 50% in this case because of the presence of extra U P M N S factor in the decays to Z/H and ν. In calculating the decay widths, as shown in appendix, we have assumed that M Z M N M h 0 .

Exclusion limits
Once the Z has been produced at a collider, it can be detected only through its decays. The leading contribution, at the LHC, accrues from qq fusion and, for moderately large |z Q |, grows as z 2 Q . Thus, a large z Q would facilitate detection and we deliberately choose to eschew this part of the parameter space, considering instead the case of moderately small z Q values, when the production cross sections are not too large. Consequently, one needs to concentrate on decay modes that are not highly suppressed. As we have seen, for much of the parameter space of interest, Z →N 3 N 3 is the dominant decay mode. The N 3 is not only very light, but also has a highly suppressed coupling to lighter species (the SM-like neutrinos) and, consequently, does not decay within the detector. Thus, this mode is not directly visible. However, with the emission of a visible particle (e.g., qg → qZ → qN 3 N 3 ), one could, instead have a signal comprising of a single jet accompanied by missing transverse momentum 11 . Although the SM backgrounds to this final state is well studied, the sheer 10 This also has immediate impact on the low-energy observables, especially those measured at the Z-peak, and is reminiscent of the B − L model as discussed, for example, in Ref. [62]. 11 Similarly, monophoton, mono-Z or mono-W signals (accompanied, in each instance by a transverse momentum imbalance) are possible too, but these suffer from additional coupling constant suppressions.
size of the background and the paucity of kinematic variables to play with renders this mode a relatively insensitive probe for a very heavy Z . As Fig. 3 shows, for such z Q , theN α N α (α = 1, 2) modes, together, can be competitive with theN 3 N 3 mode. With the N α decay branching fractions essentially being given by Eq. (7.13), a variety of final states are possible. Particularly intriguing is the possibility of the same-sign dilepton final state, that may arise when both the N 's decay into the same sign charged lepton, with the W 's subsequently decaying into, say, jets. Also possible are the trilepton plus jets and the four-lepton final states, albeit with smaller cross sections. Many of these have been studied extensively, not only in the context of many popular new physics scenarios (such as supersymmetry or theories defined in higher dimensions), but also in those similar to ours [78][79][80][81][82][83][84][85]. A simple scaling of the cross sections convinces one that the parameter space required for reproducing the neutrino masses and mixings would be accessible only once the high-luminosity version of the LHC is operational. Rather than delve into the details thereof, we concentrate instead on the most sensitive probe.

From dilepton and dijet data
Despite the relatively smaller branching fraction, the decay of the Z to a pair of charged leptons provides the strongest constraints on the parameter space, followed by the dijet signal.
Since Γ(Z ) M Z , expressing dilepton (or dijet) production in terms of an on-shell Z production followed by its decay constitutes an excellent approximation. The leading order contribution emanates from qq fusion, and has a simple structure as given in Eq. (4.1). We, though, include the next-to-leading order QCD corrections, parametrizable in terms of a K-factor of 1.3 [86]. As for the parton fluxes, we use the NNPDF2.3LO [87] parton distributions, with the natural choice for the renormalization and factorizations scales, Exclusion bounds on the model parameters, from a given experiment, can be obtained by comparing the expected signal strength with the upper bound (UB) on new physics events that the non-observation of an excess in the said experiment implies. To this end, we use the dilepton [88,89] and dijet [90,91] resonance search data that the two LHC experiments have collected (at √ s = 13 TeV) with an approximate integrated luminosity of 140 fb −1 . We start by summarizing the experimental results: • ATLAS dilepton [88]: The ATLAS collaboration has performed a high-mass spin-1 resonance search in the dilepton final state in the mass range of 0.25 TeV to 6 TeV with an integrated luminosity of 139 fb −1 . We recast their upper bound on fiducial σ × BR for a spin-1 selection with a width/mass hypothesis of 1.2% as is applicable for our analysis 12 . The definition of the fiducial phase-space region and the fiducial selection efficiency can be found in Ref. [88]. While this efficiency differs slightly for the dielectron and dimuon channels and varies, in addition, with the resonance mass, for the sake of simplicity, we use a fixed fiducial selection efficiency of 0.6 for the entire dilepton invariant mass range 13 . We obtain the observed σ × BR UB from the HepData [92] repository.
• ATLAS dijet [90]: For the dijet channel (also done with the same luminosity), the collaboration presents an UB on σ × BR × A (where the acceptance A can be approximated to 0.4). Recasting the data presented in Fig. 8a of [90] for a generic Gaussian signal in the inclusive channel with a 3% width/mass hypothesis, we obtain our exclusion limits.
• CMS dilepton [89]: The CMS collaboration has performed a high-mass spin-1 resonance search in the dilepton final state in the mass range from 0.2 TeV to 5.4 TeV. Working with an integrated luminosity ∼ 140 fb −1 , they present an upper bound on σ × BR, assuming the SM value for the width/mass ratio, namely 0.6%.
• CMS dijet [91]: For the high-mass dijet events, the collaboration uses an integrated luminosity of 137 fb −1 . We recast the observed UB on σ × BR × A (with A = 0.5) taken from Fig. 10 (lower panel) of Ref. [91] for the spin-1 resonance with a width/mass ratio of 1%. In all these four searches discussed above, we use the ones with smallest width/mass ratio which are available in those analyses as in our case Z width is much smaller compared to its mass.
As a particular example, we display, in Fig. 4, the exclusion limits on the Z mass, as gleaned from the ATLAS dilepton resonance search data, for two choices of z Q , namely z Q = −1/4 and −1/3. Working with the fiducial σ f id × BR as provided in the ATLAS paper [88], we obtain σ × BR using a fiducial selection efficiency of 0.6. The lower limits on M Z , for g z = 0.1, are about 2.2 (2.6) TeV for z Q = −1/4 (−1/3). The difference in the two exclusions can be traced to two factors, namely a slightly smaller Z -production cross section, as well as a slightly smaller branching fraction into a charged lepton pair. Note that (as promised earlier) the dependence on z Q is not too severe. Were one to be interested in the mass exclusions for other z Q and g z values, these could be obtained trivially by realizing that the production cross section scales as g 2 z , and reading off the dilepton BR from Fig. 3.

Low-energy observables
A nonzero value of z H , the U(1) z charge of the SM Higgs doublet, induces tree-level Z ↔ Z mixing. This has two main ramifications: • Tree-level contributions to the oblique parameters are induced. In particular, the tree-level contribution to the T -parameter is given by [71], (4.4) Here, M Z is the Z-boson mass in the new theory and, for our purposes, it suffices to consider the tree-level expression as given in Eq.   within the SM, is given by M 0 Z = g w v h /2 cos w at the tree-level. And, finally, α EM denotes the fine-structure constant at Z-pole. In effecting the actual calculation, the higher-order SM contributions would, of course, have to be taken into account, and we have done so. On the other hand, loop corrections to T wrought by new physics are further suppressed by large masses and the Z ↔ Z mixing angle and can be safely neglected. We use the value T = 0.07 ± 0.12 Ref. [26] in our analysis.
• A related constraint arises from the measurement of the Z-coupling of the light fermions, occasioned, again, primarily by the Z ↔ Z mixing. Determined from the forward-backward asymmetry or through the line-shape of the Z-resonance, these observables as also the Z-width [71] are very precisely measured [26].
• Another relevant constraint can come from the LEP measurements. The Z boson, despite being heavier than the LEP energies, can contribute to the e + e − →f f processes through the interference with the γ and Z mediated processes. For the sequential-SM, the 95% confidence level lower limit on the Z mass is 1760 GeV as obtained from the LEP data [93]. In our case, this limit is much more relaxed to the point of being irrelevant since the Z couplings with leptons and quarks are much smaller. than the SM-like couplings for our benchmark parameters. Therefore, we do not consider LEP constraints in our analysis.
Once again, due to the vanishing of z H for z Q = −1/3, there is no tree-level Z ↔ Z mixing, and neither of the aforementioned constraints are applicable, at least at the treelevel. The one-loop effect is too small to be of any consequence. As for z Q = −1/4, our choice commensurate with neutrino phenomenology as well as possible unification, owing to it not being too far from −1/3, the Z ↔ Z mixing is still very small, and such low-energy observables do not strongly constrain the parameter space. We show the exclusion plots in the M Z − g z plane in Figs. 5a and 5b for z Q = −1/4 and z Q = −1/3 choices respectively, while in Fig. 5c, we show the similar exclusion in the M Z − z Q plane for fixed g z = 0.15. In each case, we show exclusion regions using the latest dilepton, dijet resonance search data from the LHC, as well as those coming from the T -parameter and Z-width measurements. As expected, the dilepton data does impose severe constraints. However, owing to Br(Z →¯ ) assuming its minimum at z Q ∼ −1/4, the lower limit on M Z for both values of z Q considered here is as low as ≈ 2 TeV for g z ∼ 0.15. And while the dijet branching fraction is comparable to the dilepton one, this mode suffers from a much larger (QCD) background, and consequently the bounds are much weaker. As for T -parameter and Z-width measurements, since tree-level Z ↔ Z mixing is absent for z Q = −1/3, the constraints are virtually nonexistent, and continue to be very weak for z Q = −1/4 as well.

A Dark Matter candidate?
As we have already seen in Sec. 2 and Sec. 3, the model may have three ultralight particles, viz.
The lightest of the three, being stable, would be an apparent candidate for the dark matter. Indeed, given the lifetimes and interaction strengths, the "dark sector" could be much richer.
It is, of course, well-known that a light doublet-like neutrino would have decoupled while still relativistic and would constitute hot dark matter. Since this would have interfered with large-scale structure formation, a hot dark matter can constitute only a minor fraction of the total dark matter relic density and, fortunately, the very structure of the SM ensures that the usual neutrinos satisfy this condition. And with ν being overwhelmingly doublet-like, it too would automatically satisfy the same.
This allows us to direct our attention to the pseudoscalar A and the singlet-like neutrino N 3R . As we have already discussed, the former's mass can be trivially uplifted by the dint of adding a third singlet χ 3 such that nontrivial trilinear and/or quadrilinear terms are admissible in the scalar potential. While either of soft and hard breaking of the global symmetry-down to a single U(1) which is gauged-will render the pseudoscalar(s) massive, in the case of the former (soft breaking), the resultant mass is controllable (and we have a pNGB). This constitutes a particularly simple strategy as the neutrino masses and mixings are essentially left unchanged. As for collider signals, while the partial widths Z → Aξ i would be altered on account of the A picking up a mass, the changes are not very significant for relatively small M A values. More importantly though, the A may now decay within the detector, thereby eliminating this particular source of a missing transverse energy signal. This, however, would be replaced by more exotic (and, hence, more visible) signals at the LHC, perhaps including displaced vertices.
Concentrating on the N 3R (-like) state, let us begin by examining its mass. As a perusal of Eq. (3.4) shows, the largest contribution to this mass would accrue from the seesaw like mechanism involving the N 1R and N 2R fields. This would lead to an effective mass for the mass-eigenstate Ψ of 14 For s α3 of around 0.05 and the two heavy RHNs of the order 1.2 TeV, we get the mass of N 3 to be a few keVs. A sterile neutrino at such a scale is particularly interesting from the dark matter perspective because it can have a very long lifetime, comparable to the age of the Universe. For the keV scale DM candidates, almost the entire observed relic density can be accounted for only if m DM > 0.4 keV [94,95], which seems plausible in our case. Still, large mixings with neutrinos, while being a desirable feature from the collider search point of view, can end up producing way more dark matter than we require, thereby potentially spoiling its DM candidature. A state as light as this can only have two types of decays, namely Ψ → 3ν (where ν are the SM-like mass eigenstates) and Ψ → ν i + γ. Clearly, the latter are loop-suppressed and, hence, of relatively little significance. As for the former set of modes, it is instructive to consider, formally, the lowest-dimensional operator within the effective theory that can lead to them. This is most easily done in terms of the gauge eigenstates, viz., For a 1 ∼ O(1), this leads to which is comfortably larger than the age of the universe ( τ U ∼ 5 × 10 17 s). Naively, it might seem then that we might have as well admitted a different structure that would have allowed for a much heavier N 3R . This, however, is misleading. As we have seen in Sec.3, the diagonalization of the mass matrix leads to mixings between the ν-like and the N 3 like eigenstates. In other words, the mass eigenstates Ψ, ν are symbolically given by where generational dependence has been omitted. Other mixings connecting ν i , N 3R with N 1,2 are not taken into account as they are much suppressed and, therefore, not significant for the issue at hand.
Owing to the fact that the N 3R is a singlet, this mixing immediately leads to a ZΨν coupling which, in turn, leads to a Z-mediated contribution to the Ψ → 3ν decay. The corresponding partial widths are leading to a lifetime τ Ψ > ∼ 10 18 s, for m N ∼ 6 keV and mixing of O(10 −2 ). So for such masses and mixings, we still can barely have τ Ψ > τ U . This ostensible enhancement compared to the earlier estimate can be traced back to the fact that several powers of (x/Λ) in the aforementioned effective operator are actually subsumed in the suppression of m N 3 itself, and must not be double counted. At this point, several issues need to be delved upon. While it might seem that we naturally have τ Ψ > τ U , in reality, τ Ψ depends crucially on the values of the Wilson coefficients, and, thus, such a requirement on the lifetime imposes conditions on the WCs. It should be appreciated though that, for such a light DM particle, τ Ψ > τ U is not a strict requirement. In fact, a τ Ψ value even somewhat smaller than τ U may also be admissible for this would only mean that a fraction of the DM has decayed in the course of the Universe's evolution yet leaving behind sufficient relic density. Since the decay is into neutrinos alone, the only discernible effect would be through altering the neutrino-photon ratio in the early universe, thereby altering the effective number of relativistic degrees of freedom, and thus invite constraints from this measurement. However, the existence of such a restriction depends crucially not only on the epoch of this decay, but also on specific mechanism (e.g., freeze-in versus freeze-out) of DM relic density generation, and is not immediately applicable to the case at hand.
On the other hand, even if a τ Ψ value somewhat smaller than τ U is obtained, these same operators would lead, at one loop, to Ψ → ν i +γ manifesting itself through X-ray lines. The non-observation of such signal in different low energy experiments, mandates [96,97] that the sterile-SM neutrino mixing angle satisfies θ i 10 −2 if a 6 keV Ψ were to provide full DM relic density. The simplest way to satisfy such seemingly incompatible constraints would then be to assume that the Ψ provides for only a small fraction of the relic density in the form of a potentially warm component, especially since for certain regions of the parameter space, it could be produced non-thermally. More interestingly, the Ψ has considerable selfinteraction (mediated by the Z ) with a suppressed, but long-distance component mediated by the pseudoscalar A. This has the potential to provide some pressure to the DM fluid and, thereby, address certain long-standing issues pertaining to details of structure formation.
To anoint Ψ to be the main or even a significant DM constituent, one must ascertain not only whether the correct relic density can be reached but also whether the scenario falls foul of other constraints, both cosmological as well as those emanating from laboratory tests (both direct and indirect detection). This demands detailed analysis that is beyond the scope of the present work. However, at the same time, we want to emphasize a few general issues. In our quantitative analysis we made several simplifying assumptions regarding x i , Λ and some of the WCs. Tweaking these assumptions can substantially change the masses and mixings of the neutrino, as is seen in the context of a Frogatt-Nielsen scenario through the introduction of different scales through different powers of a scaling factor [98]. In a similar vein, by altering the U(1) z charges of the N i (while maintaining anomaly cancellation) and/or the scalar fields χ a , the neutrino mass matrix can be changed. This would allow us a much larger mass for the N 3 -like state, viz. O(MeV), with further suppressed N 3 -ν i mixings thereby still allowing for τ Ψ > ∼ τ U . Consequently, the standard freeze-in mechanism would hold for such a DM. We would like to postpone these issues to a future project.
To examine the falsifiability of our hypothesis, it is important to consider the strength of the interactions that mediate low-energy scattering involving Ψ and the SM particles. Fortunately, such interactions are not unduly suppressed thanks to the fact that both N 3R and the SM fermion carry U(1) z quantum numbers. Consequently, the interaction strength is governed only by g 2 z /M 2 Z or, equivalently, by x −2 i , and in the case of freeze in DM generation, relic density is proportional to this interaction, that, M Z being at the TeV-scale, exactly represents an example of how entire relic density can be reproduced. Rather than present a full analysis, we refer the reader to the existing literature. For example, it has been shown, in an analogous context, in Ref. [99] that a parameter space, consistent simultaneously with the requisite relic density, the measurement of the cosmologically relevant effective relativistic degrees of freedom and energy injection, from DM annihilation, into the cosmic microwave background radiation, can be found. The required interaction strength is of the same order as what transpires naturally in our model and would leave such particles undetectable in the currently operative (satellite-based) indirect detection experiments [99]. Even more interestingly, such a DM is likely to be detectable not only at the next generation of direct detection experiments, but also at the Super-Belle detector [100].
Before we end this section, we would like to remind the reader of a possibility that we did not elaborate on. Consider, for example, the case where the extra global U(1) is not broken by terms in the potential. The Yukawa couplings, nonetheless, do break it and, consequently, quantum corrections would lift the mass of the Goldstone by a tiny amount, leaving it stable on cosmological time scales. Free from restrictions (such as those imposed by X-ray or Lyman-α observations), this could, again, play a significant role in the evolution of galaxy clusters etc. [101]. A detailed examination of such effects is beyond the scope of this paper and is postponed for a future study.

Summary and Conclusion
With the aim of explaining neutrino masses without invoking either ultrasmall Yukawa couplings or an almost inaccessible new (seesaw) scale, we consider a scenario where the gauge symmetry has been augmented by an extra U(1) z . If its action on the SM particles is nontrivial, but generation-invariant (so as to allow for a single SM Higgs to give masses to the charged leptons), then the possible charge assignment for the right-handed neutrinos (RHN) is severely restricted by the requirement of gauge (and mixed gauge-gravity) anomaly cancellation. (We assume here that, unlike in certain popular schemes such as the inverse seesaw mechanism, we have the minimum possible number of RHNs.) Only the most trivial such assignment allows for tree-level neutrino Dirac mass terms. On the other hand, bare Majorana mass terms cannot be incorporated. Indeed, analogues of the Weinberg term can be written only if the new Higgs breaking the U(1) z have one of two specific choices of the charge.
For any choice of the RHN charges other than the most trivial one, not only are renormalizable Dirac mass terms disallowed, but so are the Majorana mass terms except for specific choices of the U(1) z breaking Higgs bosons. Completely unrelated to this, the absence of any resonance in the LHC data has pushed the mass of the new gauge boson Z to above several TeVs.
In view of this, we assume an agnostic standpoint claiming that any such theory can, at best, be the low-energy limit of a more fundamental theory, characterised by a cut-off scale Λ. This, immediately, allows us to write non-renormalizable terms suppressed by powers of Λ. While a wide variety of such terms, in principle, can be written, we concern ourselves only with the neutrino sector. Invoking the next to the trivial quantum number assignment for the RHNs, we then write down all relevant higher-dimensional termsá la the Froggatt-Nielsen mechanism. Using the power of higher dimensional operators to the hilt, we generate tiny neutrino masses without any need to invoke tiny Yukawa couplings. Indeed, even without using all the free parameters of the theory, it can naturally reproduce the experimentally observed neutrino mixings and mass-squared differences, while satisfying the cosmological bound on the sum of masses as well as that from non-observation of neutrinoless double-beta-decay. Simultaneously, it prophesies, amongst others • a pair of heavy RHNs N 1,2 at the 1 TeV mass scale that decay promptly into W , νZ and promise interesting signals at the high-luminosity run of the LHC; • a moderately heavy Z (m Z < ∼ 3 TeV for g Z ∼ 0.15) that escapes LHC boundsfrom dilepton and dijet searches-despite having unsuppressed couplings with the quarks and leptons, simply by virtue of decaying primarily into the RHNs. Similarly, for natural choices of U(1) z charges (especially those commensurate with possible charge quantization), the LEP constraints such as those on the oblique parameters are trivially satisfied; • a light RHN N 3 in the keV-MeV range. With the Z having a large branching fraction into a N 3 -pair, and with the N 3 being stable at the collider timescales, this would lead to additional contribution to the monojet (monophoton) plus missing transverse momentum signal at the LHC; Indeed, for a large part of the parameter space, the N 3 can have a lifetime comparable to or even greater than that of the Universe and, thus, can constitute a warm DM component.
• a pseudoscalar pseudo-Nambu Goldstone boson, with its mass uplifted only by quantum corrections or additional soft terms in the scalar potential (the latter being absent in the simplest realization). This has the potential of being an additional contributor to the DM relic density (while escaping many of the constraints applicable to N 3 ). Furthermore not only does it have non-negligible self-interaction, but it can also mediate N 3 scattering thereby playing an important in not only determining the relic density, but also in engendering a non-negligible pressure term for the DM fluid and thereby affecting the details of structure formation.
The model presented, thus, offers much more than an understanding neutrino phenomenology. Not only does it offer tantalizing prospects at the LHC, but also intriguing avenues to explore in the context of dark matter and details of structure formation. We hope to return to more in-depth study of these issues in a future publication.

Decay widths of Z
In this Appendix, we provide the analytical expressions of the tree-level partial widths of various two-body decay modes of Z . These expressions are computed using the Feynman rules obtained from the interaction Lagrangian shown before them.
• Z →f f : For the following interaction Lagrangian, the expression for the Z →f f partial width is given by In the above, g L and g R are the left-and the right-handed couplings respectively, M f is the mass of the fermion f and N c is the corresponding number of colors.
• Z → ν R ν R : For the following interaction Lagrangian, the expression for the Z → ν R ν R partial width is given by where g ν is the coupling and M ν R is the mass of the RHN.
• Z → W + W − : For the following triple gauge boson interaction with strength λ W , the expression for the Z → W + W − partial width is given by (7.6) • Z → ZS: From the following interaction with dimensionful coupling strength µ S , where S is a CP even scalar, the expression for the corresponding partial width is given by (7.8) • Z → SA: From the following interaction with cubic coupling L Z SA = g p Z µ ∂ µ SA , (7.9) where S is a CP -even scalar and A is a CP -odd scalar 3/2 (7.10)

Decay widths of heavy RHN
The heavy RHN decay modes are given by the Lagrangian of the form: The decay rates are then given by: where U νν is approximately the P M N S matrix. U νN is the mixing between the light SM neutrinos and the heavy RHNs given by [102] [103] U νN = −D(M diag N ) −1 . (7.14)