Type-I Seesaw with eV-Scale Neutrinos

We consider seesaw type-I models including at least one (mostly-)sterile neutrino with mass at the eV scale. Three distinct situations are found, where the presence of light extra neutrinos is naturally justified by an approximately conserved lepton number symmetry. To analyse these scenarios consistently, it is crucial to employ an exact parametrisation of the full mixing matrix. We provide additional exact results, including generalised versions of the seesaw relation and of the Casas-Ibarra parametrisation, valid for every scale of seesaw. We find that the existence of a light sterile neutrino imposes an upper bound on the lightest neutrino mass. We further assess the impact of light sterile states on short- and long-baseline neutrino oscillation experiments, emphasise future detection prospects, and address CP Violation in this framework via the analysis of CP asymmetries and construction of weak basis invariants. The proposed models can accommodate enough active-sterile mixing to play a role in the explanation of short-baseline anomalies.


Introduction
Most of the present data on Neutrino Physics are consistent with the hypothesis of having only three active neutrinos. Nevertheless, there is a small subset of experiments which seem to require the presence of New Physics (NP). The first indication hinting at the presence of NP was provided by an excess in the results of the LSND experiment, where electron anti-neutrinos were observed in a pure muon anti-neutrino beam [1,2]. One of the simplest explanations of the LSND result involves the existence of an anti-neutrino with a mass-squared difference ∆m 2 of about 1 eV 2 . Taking into account that ∆m 2 atm is of order 10 −3 eV 2 and ∆m 2 solar of order 10 −4 eV 2 one concludes that the LSND result would require a fourth neutrino. On the other hand, the invisible decay of the Z gauge boson shows that there are only three active neutrinos with a mass less than a half of the Z mass [3], implying that if a fourth light neutrino exists it must be sterile, i.e., a singlet under the gauge symmetry of the Standard Model (SM). The existence of extra (sterile) neutrinos should then be reconciled with cosmological constraints, which call for a suppressed thermalisation of these massive neutrinos in the early Universe, given the effective neutrino number N eff = 2.99 +0. 34 −0.33 (95% CL, from TT,TE,EE+lowE+lensing+BAO) measured by Planck [4] 1 .
Meanwhile, new anomalies have appeared in Neutrino Physics supporting the hypothesis of the existence of light sterile neutrinos. The indications for the existence of a sterile neutrino of mass of order 1 eV come from short-baseline (SBL) neutrino oscillation experiments. They started with the LSND result in the nineties. At that time, this result was not confirmed by KARMEN [7]. However, KARMEN had a shorter baseline than LSND and therefore could not exclude the whole parameter space available to LSND. This was followed by the MiniBooNE experiment [8] with inconclusive results 2 . Recently, new interest in the LSND result was sparked by the"reactor anti-neutrino anomaly" due to a deficit of the number of anti-neutrinos observed in several different reactor neutrino experiments, when compared with the theoretical flux calculations [12][13][14]. A crucial and independent development has been provided by the DANSS [15] and NEOS [16] collaborations, whose programmes include comparing spectra at different distances from the anti-neutrino source. The preferred fit regions of these independent experiments interestingly overlap near ∆m 2 ∼ 1.4 eV 2 and sin 2 2ϑ 14 ∼ 0.05, with ϑ 14 being an effective mixing angle as interpreted in a 3+1 scheme. Also of relevance is the so-called "Gallium neutrino anomaly", discovered in 2005-2006 [17][18][19], albeit of less significance. For recent reviews on eV-scale sterile neutrinos and additional references, see [20,21].
The purpose of this paper is to investigate the possibility of obtaining in a natural way at least one sterile neutrino with a mass of order eV in the framework of the general type-I seesaw mechanism [22][23][24][25][26]. The crucial point is that we shall consider a special case of the seesaw framework. Instead of having three heavy sterile neutrinos, as in the usual setup, at least one of the sterile neutrinos should be light while, at the same time, its mixing with the light active neutrinos should be small enough to comply with existing experimental bounds, but large enough to be relevant to low energy phenomenology. Two important challenges are to find solutions that are stable under renormalisation, and to inquire if these spectra, with at least one neutrino with a mass of order eV, might indeed explain the SBL anomalies.
For definiteness, let us recall how the conventional seesaw mechanism works. It consists of an extension of the SM where three right-handed neutrinos are added to the standard spectrum. As a result, the neutrino mass terms include a Dirac mass matrix, denoted m, generated by the breakdown of the electroweak (EW) symmetry, and a Majorana mass term, denoted M, with the scale of M much larger than the scale of m. In general this leads to three light neutrinos with masses of order m 2 /M and three heavy neutrinos with masses of order M. The generic seesaw framework leads to an active-sterile mixing of order m/M, too small to be of relevance to low energy physics, while providing a framework for leptogenesis [27]. In the derivation of the standard seesaw formulae, one performs a block diagonalisation of the 6 × 6 complex neutrino mass matrix, obtaining approximate relations that are valid to an excellent approximation. Some of the approximate formulae no longer hold in the special cases which we are considering. However, there are important exact relations which continue to be valid in our case. We find viable models with at least one sterile neutrino with a mass of order eV by imposing a U(1) symmetry (see e.g. [28]) allowing for small breaking terms. Before the breaking, for special assignments of leptonic charges, the lightest neutrinos are naturally massless at tree level, acquiring calculable small masses after the breaking and complying with the experimental ∆m 2 values after radiative corrections.
The paper is organised as follows. In the next section, we describe our setup, settle the notation and present a useful parametrisation of the mixing matrix as well as some exact results concerning the Dirac mass matrix, neutrino masses and deviations from unitarity. In section 3 we discuss the size of such deviations from unitarity in the 3×3 leptonic mixing matrix. In section 4 we describe how one-loop mass corrections can be controlled within the considered framework. In section 5 we present explicit numeric examples and go through their phenomenology, while section 6 is dedicated to the study of CP Violation within the type-I seesaw, with emphasis on CP Violation measurements and CP-odd weak basis invariants. Finally our conclusions are presented in section 7.

Framework
We work under the type-I seesaw framework, in a model with three right-handed neutrinos added to the SM. The leptonic mass terms are given by: where n 0 L = (ν 0 L , C ν 0 R T ) T and the zero superscript denotes a general flavour basis. Without loss of generality, one may choose a weak basis where m l is real and diagonal. The analysis that follows is performed in this basis, meaning ν 0 L = (ν eL , ν µL , ν τL ). The neutrino mass matrix M is a 6 × 6 complex symmetric matrix and has the form: This matrix is diagonalised by the unitary transformation where D is diagonal real non-negative and contains all neutrino masses, Here, d contains the masses of the three known light neutrinos, d = diag(m 1 , m 2 , m 3 ), and D the masses of other neutrinos, D = diag(M 1 , M 2 , M 3 ). The 6 × 6 unitary matrix V can be written where K, R, S and Z are 3×3 matrices. Using the unitarity of V , one can obtain [29] a series of exact relations relating the matrices K, R, S, and Z, examples of which are KK † + RR † = 1 and KS † + RZ † = 0. We shall show that in order to study deviations of unitarity, it is useful to parametrise V in a different way.

A Novel Parametrisation for the Leptonic Mixing Matrix
In Ref. [29] we introduced an especially useful parametrisation of the 6 × 6 leptonic mixing matrix that enables to control all deviations from unitarity through a single 3 × 3 matrix which connects the mixing of the active and sterile neutrinos in the context of type I seesaw. It was written: where it is assumed that K and Z are non-singular. From the aforementioned unitarity relation KS † + RZ † = 0 one promptly concludes that Thus, a generic 6 × 6 unitary matrix V , in fact, only contains three effective 3 × 3 matrices K, Z and X. Furthermore, from the same unitarity of V and from the singular value decomposition X = W d X U † , one finds that K and Z can be written as: where 3 U K ,W Z , U and W are all 3 × 3 unitary matrices, d X is a diagonal matrix with real non--negative entries, and we have defined an additional unitary matrix V ≡ U K U † . The matrices U and W diagonalise the Hermitian products X † X and XX † , respectively: Any unitary matrix to the left of Z -like the product W Z W † in Eq. (2.8) -is unphysical as it can be rotated away via a weak basis transformation which does not affect the form of m l . Accordingly, one can choose to work in a weak basis for which Σ = 1 in the general expression with Σ unitary. Note, however, that Σ = 1 in the numerical 'symmetry' bases considered later on in sections 4 and 5.
The matrix K plays the role of the PMNS mixing matrix, as it connects the flavour eigenstates ν αL (α = e, µ, τ) to the lightest mass eigenstates. From Eq. (2.8), it is clear that K is unitary if and only if d 2 X = 0. Thus, the deviations from unitarity are manifestly expressed in the diagonal matrix d 2 X containing the (squared) singular values of X.
In summary, a generic 6 × 6 mixing unitary matrix V can be simplified and be written in terms of just one 3 × 3 unitary matrix V and of explicit deviations from unitarity, parametrised by a 3 × 3 matrix X: i.e.
In general, there are no restrictions on the matrix X. However, in a type-I seesaw model, the mixing matrix V must also obey the mass relation stated in Eq. (2.3), and the 6 × 6 neutrino mass matrix M is not general: some entries are zero at tree level. This imposes a restriction 4 on X, d + X T D X = 0 , (2.13) which implies that it is possible to write X as: Since O c is an orthogonal complex matrix, not all of its elements need to be small. Furthermore, not all the M i need to be much larger than the electroweak scale, in order for the seesaw Total   Moduli  3  3  3  3  3  15 = 9 + 6   Phases  −3  0  0  3  6  6   Table 1: Physical parameter counting in type-I seesaw with three sterile neutrinos. The 15 moduli correspond to 9 lepton masses (3 charged-lepton masses and 6 neutrino masses) and to 6 mixing angles. There are 6 physical phases, as rephasing the charged leptons can remove 3 phases from V . Recall that m l is real and diagonal in the considered weak basis.
mechanism to lead to naturally suppressed neutrino masses. These observations about the size of the elements of X are especially relevant in view of the fact that some of the important physical implications of the seesaw model depend crucially on X. In particular, the deviations of 3 × 3 unitarity are controlled by X, as shown in Eq. (2.8). On the other hand, from Eq. (2.13) one can also see that X must not vanish, in order to account for the non-zero light neutrino masses.
From the above, one concludes that the set {m l , d, D,V, O c } of matrices is sufficient to describe lepton masses and mixing at tree level. In the working weak basis, there are 9 lepton masses in the first three matrices, while mixing is parametrised by 6 angles and 6 CP-violating (CPV) phases, contained in the unitary matrix V and in the orthogonal deviation matrix O c . Parameter counting is summarised in Table 1 and is in agreement with, e.g., Refs. [35][36][37]. Coincidentally, these numbers of angles and CPV phases match those of a general 3+1 scenario (see e.g. [38]), even though three right-handed neutrinos have been added to the SM. This is a consequence of having a type-I seesaw UV completion, which requires the zero block in Eq. (2.2).
In this paper, we consider the possibility of having at least one sterile neutrino with a mass of order eV arising from the seesaw mechanism in a model with three right-handed neutrinos added to the SM. We analyse the different aspects and consequences of the phenomenology of such a model. With this aim, relations between observables and parameters which are independent of the seesaw limit are derived in the following subsection.

Exact Relations at Tree Level
Recall that, in our working weak basis, m l is diagonal and K is directly identified with the non-unitary PMNS matrix. Moreover, K and Z take the forms given in Eq. (2.11) and one has: This exact formula is to be contrasted with the known parametrisation for the neutrino Dirac mass matrix developed by Casas and Ibarra [39], which is valid in the standard seesaw limit of M m and reads (2.18) in the weak basis where m l and M = diag(M 1 ,M 2 ,M 3 ) ≡D are diagonal. Here, O CI c is an orthogonal complex matrix and U PMNS represents the approximately unitary lepton mixing matrix. In this limit of M m, the light neutrino mass matrix m ν can be approximated by: As a side note, let us remark that it is possible to obtain a parametrisation for m which is exact and holds in a general weak basis by following the Casas-Ibarra procedure. One finds: where once againÕ CI c is a complex symmetric matrix. However,d andD do not contain physical masses, but are instead diagonal matrices with non-negative entries obtained from the Takagi decompositions At this stage, one may wonder whether there exists an exact relation, analogous to Eq. (2.19) which is valid in any region of parameter space. One can actually deduce such a relation for an arbitrary number of active and sterile neutrinos. Consider the following decomposition of a block-diagonal matrix: where A, B, C, and D are complex p × p, p × q, q × p, and q × q matrices, respectively, and one has assumed that D is non-singular. From this it follows that with m i (i = 1, . . . , p) and M j ( j = 1, . . . , q) denoting the neutrino masses. For the case of interest, p = q = 3 and one has: We stress that these relations are exact and that no assumptions have been made about the relative sizes of the m i and M j . It is clear from Eq. (2.25) that the smallness of neutrino masses in this framework may have its origin in the largeness of the M j (with respect to the EW scale), or in the suppression of | det m| due to e.g. an approximate symmetry.
In the framework of the type-I seesaw, it is the block K of the matrix V that takes the role played by the U PMNS matrix at low energies, typically taken as unitary and parametrised accordingly (see e.g. the standard parametrisation [3]). Clearly, in this framework, K is no longer a unitary matrix. When considering the deviations from unitarity of K, one must comply with experimental bounds, while at the same time investigate whether it is possible to obtain deviations that are sizeable enough to be detected experimentally in the near future. Using the above parametrisation, this translates into making appropriate choices for the matrix X. Deviations from unitarity of K can be parametrised as the product of an Hermitian matrix by a unitary matrix [44]: where η is an Hermitian matrix. In the previous section, we have instead parametrised K with an Hermitian matrix to the right and the unitary matrix V to the left, see Eq. (2.11). These right-and left-polar decompositions are unique since we are dealing with a non-singular K by assumption. Moreover, they can be connected explicitly: Expanding in powers of X (or equivalently of d X ), one obtains Constraints on the entries of η depend on the mass scale of the new neutrinos. Bounds on η can be found in the literature for the scenario in which all three heavier neutrinos have masses above the EW scale [43,44]. As pointed out in [44], in such a case it is very useful to parametrise K with the unitary matrix on the right, due to the fact that, experimentally, it is not possible to determine which physical light neutrino is produced. Therefore, one must sum over the massive neutrino fields and observables depend on KK † . From the unitarity relation KK † + RR † = 1 and Eq. (3.1), one has i.e. there is a straightforward connection between KK † , RR † and the deviations from unitarity, expressed in η.
When one has one or more light sterile neutrinos, the aforementioned bounds cannot be directly applied, as some states are kinematically accessible and different sets of experimental constraints need to be taken into account, depending on the spectrum at hand. In this case, observables can constrain directly the entries of R, and not just the product RR † . For light sterile neutrinos with eV-scale masses, the most stringent bounds on deviations from unitarity come from oscillation experiments [45], such as BUGEY-3 [46], MINOS [47], NOMAD [48,49] and Super-Kamiokande [50]. In our analysis, the relevant exclusion curves in the sin 2 2ϑ αβ -∆m 2 planes (see section 5) are considered and translated into constraints on the elements of the mixing matrix block R. If one is dealing instead with keV or GeV -TeV sterile neutrinos, it is important to take into account the experimental bounds coming from β-decay experiments (see e.g. [51] and references within) and from LHC searches for heavy Majorana neutrinos [52][53][54].
Another crucial experimental input, also taken into account in our analysis, is the limit on the µ → eγ branching ratio obtained by the MEG Collaboration, BR(µ → eγ) < 4.2 × 10 −13 (90% CL) [55], one of the most stringent bounds on lepton flavour violating processes. This bound is expected to be relevant whenever the heavier neutrino masses are around or above the EW scale, as a GIM cancellation arises for lighter states (see for instance Eq. (40) of Ref. [44]).

Restrictions on the Neutrino Mass Spectrum
The type-I seesaw model that we consider here, with at least one sterile neutrino with a mass around 1 eV, also leads to some restrictions on the light neutrino mass spectrum at tree level. In particular, we find an upper bound on the mass m min of the lightest neutrino, as a function of the deviations from unitarity.
Taking into account the parametrisation (2.14) for the matrix X controlling deviations from unitarity, and for eigenvalues d 2 From this, and recalling that and conclude that where naturally M 1 ≤ M 2 ≤ M 3 and m min = m 1 (m 3 ) for normal (inverted) ordering. Then, inserting the inequality ∑ i O c 1i 2 ≥ 1, valid for any orthogonal complex matrix, we find As discussed, when one has one or more light sterile neutrinos, the typical stringent conditions on the deviations from unitarity do not apply. Thus, one may consider larger deviations from unitarity, even of the order of the smallest U PMNS angle, i.e. O(0.1) [45]. Since in the scenarios of interest the lightest of the heaviest neutrinos has a mass of M 1 ∼ 1 eV, using Eq. (3.8) we find a bound for the mass of the lightest neutrino: Note that this bound becomes stronger as one considers smaller and smaller deviations from unitarity. Taking into account the measured light neutrino mass-squared differences, we conclude that the light neutrinos cannot have masses above O(0.1) eV under these conditions, a statement which is also supported by cosmological bounds [56].

Neutrino Oscillations
In the presence of deviations from unitarity, neutrino oscillation probabilities are modified [41,45]. If n of the heavier neutrinos are accessible at oscillation experiments, then a 3 × (3 + n) submatrix Θ of V enters the computation of oscillation probabilities, where R 3×n contains the first n columns of R. For a given experimental setup, and depending on their masses, the heavier states may already be produced incoherently or instead lose coherence before reaching the detector, due to wave-packet separation (see e.g. [57]). The probability of transition between flavour (anti-)neutrinos ν ( -) α and ν ( -) β , or of survival for a given flavour (α = β), with α, β = e, µ, τ, can be shown to take the form where the plus or minus sign in the second line refers to neutrinos or anti-neutrinos, respectively. Here, L denotes the source-detector distance, E is the (anti-)neutrino energy, and one has defined Note that if n = 3 then ΘΘ † = KK † + RR † = 1 3×3 due to the unitarity of the full 6 × 6 mixing matrix V and Eq. (3.11) reduces to the usual unitary formula. It should be pointed out that the normalisation (ΘΘ † ) αα (ΘΘ † ) ββ in (3.11) will cancel in the experimental event rates, due to similar correction factors appearing in production rates and detection cross-sections [41,57]. Nevertheless, we explicitly keep it in subsequent expressions. It will turn out to be negligibly close to unity for our particular numerical examples. The term proportional to |(ΘΘ † ) αβ | 2 is instead known to correspond to a "zero-distance" effect [41,58]. It will also turn out to be negligible for our explicit numerical examples.
In what follows, we will consider approximate forms of Eq. (3.11), having in mind SBL and long-baseline (LBL) experimental setups. Since LBL experiments realistically need to take matter effects into account, our formulae in those cases are simply indicative.
4 Structure of the Mass Matrix

One-loop Corrections
So far we have focused on neutrino masses and mixing at tree level. However, in general, one expects one-loop corrections δM L to the 0 (3×3) block of M in Eq. (2.2). As these are not guaranteed to be negligible, one should keep track of them in order to properly scan the parameter space of seesaw models. They are inherently finite and are given by [59,60] (see also [61]): where δM Z L and δM H L represent contributions depending on the Z and Higgs boson masses, m Z and m H , respectively. Explicitly, one has (see also Appendix A of Ref. [60]): in a generic weak basis, with v 174 GeV being the Higgs VEV and with D, K and R given in Eqs. (2.4) and (2.11). This result can be cast in a simple form: where naturally f is applied element-wise to diagonal matrices, with Models with very small deviations from unitarity (standard seesaw) have a very small X and hence a correspondingly small R = KX † . For these, the one-loop δM L corrections are negligible, as can be seen from Eq. (4.3). Namely (aside from the loop-factor suppression), the terms with K are suppressed by the light neutrino masses d, whereas the effect of the heavier neutrino masses in D is regulated by the small entries of R. However, in models with sizeable deviations from unitarity, R is not small and controlling δM L requires a mechanism such as a symmetry at the Lagrangian level.

Approximately Conserved Lepton Number
Relatively light sterile neutrinos can arise naturally in a seesaw framework in the presence of an approximately conserved lepton number [62][63][64]. Such a U(1) L symmetry, when exact, imposes specific textures on the mass matrices m and M. These textures may be slightly perturbed when the symmetry is approximate, 5 allowing for non-vanishing Majorana neutrino masses and non-trivial mixing.
We are interested in scenarios where at least one of the mostly-sterile neutrinos is light, with a mass of O(eV), in order to establish a connection to the SBL anomalies. We are further looking for situations where some of the Yukawa couplings are of order one. The choice of lepton charges should then be such that, in the exact conservation limit: i) M has zero determinant, 6 and ii) not all entries of m are small. These conditions limit the possible U(1) L charge assignments.
For this case, the only sensible choice for the doublet charges is λ L = (0, 0, 0). The mass matrices in the symmetric limit read: Breaking the symmetry will generate the light neutrino masses, two (mostly-)sterile states with masses M 1 and M 2 that can be much smaller than M 3 , and a heavy sterile with a mass close to M 3 . As expected, some Yukawa couplings remain of O(1), which can also be understood from Eq. (2.16), expressing the dependence of the Dirac mass matrix m on the sterile masses contained in D. This case is further separated into two subcases: one can allow for a hierarchy M 2 M 1 (case Ia), which may arise in a scenario of stepwise symmetry breaking, or instead focus on a single new light-sterile scale, with M 1 ∼ M 2 (case Ib).

Case II
For this case, one is instead led to λ L = (1, 1, 1). In the exact conservation limit, the mass matrices are given by: (4.6) In this limit, one has two degenerate neutrinos with mass |A| 2 + |B| 2 and opposite CP parities, forming a single heavy Dirac particle. Breaking the symmetry will allow for the generation of light neutrino masses and for another massive sterile state to arise, with a mass than can be much smaller than |A| and |B|. It will additionally lift the mass degeneracy for the Dirac neutrino, producing a pseudo-Dirac neutrino pair [67,69]. As pointed out in [70], a strong mass degeneracy translates into a symmetry in the R block of the mixing matrix, namely R α2 ±i R α3 (α = e, µ, τ). Such a relation can be seen to play a fundamental role in suppressing the effect of the large masses M 2 and M 3 in the one-loop correction δM L , see Eq. (4.3). It signals that one is close to the limit of lepton number conservation, even if R α2 and R α3 are not extremely suppressed. 7 One is then allowed to have relatively large Yukawa couplings even if M 2 M 3 are not as large as the M 3 of case I. This can be seen from Eq. (2.16), which can be written in the form m = R D Z −1 * . The mass of the pseudo-Dirac pair can be at the TeV scale [64,[70][71][72][73], since the size of the lightest neutrino masses is protected by approximate lepton number conservation. The same symmetry and effects are present in the examples given in Ref. [29].
In the following section, we perform a numerical analysis focusing on cases Ia, Ib and II and incorporating an eV sterile neutrino in the seesaw spectrum while allowing for a mixing matrix K with sizeable deviations from unitarity.

Numerical Analysis and Benchmarks
For each of the cases Ia, Ib and II defined in the previous section, we explicitly provide a numerical benchmark for the seesaw mass matrices, and explore the parameter space of qualitatively similar seesaw structures. As anticipated in section 3.2, we further provide approximate forms of the transition probabilities of muon to electron (anti-)neutrinos, P ν Eq. (3.11) while having in mind SBL and LBL setups, for each of the three scenarios. Given that recent global fits [74,75] disfavour a light neutrino mass spectrum with inverted ordering (IO) with respect to one with normal ordering (NO) at more than the 3σ level, we restrict the mass ordering to NO in our numerical examples.
Before proceeding, note that the three scenarios of interest exhibit some correspondence to the commonly considered 3+1+1 (case Ia), 3+2 (case Ib), and 3+1 (case II) schemes, see for instance [20]. Thus, even though the connection to the latter is not exact -in particular, the spectrum of case Ib is not that of a typical 3+2 scenario -it may prove useful to consider quantities therein defined in our analysis, namely [76] sin 2 2ϑ  µe ∈ [6.5 × 10 −4 , 2.6 × 10 −3 ] (99.7% CL) in the 3+1 scheme. This result may also be of relevance in the 3+1+1 scheme. Although we take these intervals as guidelines in our numerical explorations, it is not our aim to address the tensions in the current experimental 7 Nonetheless, it is true that in the exact conservation limit d = X = R = 0. situation of the SBL anomalies. Thus, we only restrict our sterile neutrino parameter space at the outset through the conservative bounds ∑ i |R αi | 2 < 0.1 (α = e, µ, τ), and via the constraints of [48,49,51] on mixing matrix elements corresponding to large mass-squared differences ∆m 2 ∼ 10 eV 2 − 1 keV 2 , as anticipated in section 3.

Case
The numerical data for the benchmark corresponding to this case is given in Table 2a, where the one-loop correction of Eq. (4.3) has been taken into account. Apart from the three light mostly--active neutrinos, the spectrum includes three mostly-sterile neutrinos with masses M 1 ∼ 1 eV, M 2 ∼ 1 keV, and M 3 a few orders of magnitude below the grand unification (GUT) scale, M 3 ∼ 10 14 GeV. The keV-scale neutrino may be a viable dark matter candidate [51,77].
For the spectrum of case Ia, one has n = 2 in Eq. (3.10). In the context of a LBL experiment (e.g. DUNE [78]), the expression of Eq. (3.11) applied to the transition probability of muon to electron (anti-)neutrinos can, in this case, be approximated by: where terms depending on ∆ 4 j , ∆ 5 j 1 have been replaced by their averaged versions (sin 2 ∆ i j → 1/2, sin 2∆ i j → 0). While the normalisation and the first term in this equation signal the loss of unitarity and a zero-distance effect, respectively, the last two terms explicitly represent the effects of the two lightest mostly-sterile states in oscillations. If one is in a condition similar to that of the numerical benchmark of Table 2a, for which |(ΘΘ † ) µµ (ΘΘ † ) ee − 1| and |(ΘΘ † ) µe | 2 are negligible, this expression can be further approximated by: where we have defined a 3ν-framework transition probability which, however, incorporates the effects of deviations of K from unitarity, and have used the definition of Eq. (5.1), the unitarity of the full 6 × 6 mixing matrix, and the fact that In a SBL experiment (e.g. MicroBooNE [79]), the relevant form of Eq.
Θ µ j Θ * e j sin 2∆ 41 , (5.5) with ∆ 41 ∆ 42 ∆ 43 , and where terms depending on ∆ 5 j 1 have been replaced by their averaged versions (sin 2 ∆ 5 j → 1/2, sin 2∆ 5 j → 0). In this context, one is sensitive to oscillations due to the scale of the mass-squared differences ∆m 2 4 j with j = 1, 2, 3, while the oscillations pertaining to smaller mass-squared differences have not yet had a chance to develop. Finally, if one is in a condition similar to that of the numerical benchmark, this expression can be simply approximated by: where once again one has taken into account the unitarity of the full mixing matrix and the fact that |R α1 | 2 |R α2 | 2 ≫ |R α3 | 2 .
To further explore the parameter space of case Ia, we have produced numerical seesaw structures by specifying tree-level values of the unitary part V of the mixing matrix K, the mostly-active and mostly-sterile masses in d and D, and by scanning the complex orthogonal matrix O c , parametrised as a product of three complex rotations times a sign corresponding to its determinant. We are interested in seesaw structures qualitatively similar to our benchmark, so that we specify (at tree level) M 1 = 1 eV and M 2 = 1 keV, while considering three different values for the heaviest neutrino mass, M 3 ∈ {10 13 , 10 14 , 5 × 10 14 } GeV. While the lightest neutrino mass m min is scanned in the range [10 −4 , 0.1] eV, the remaining elements of d are fixed by specifying the solar and atmospheric mass differences. The 3ν mixing angles and Dirac CPV phase entering V as well as the aforementioned 3ν mass-squared differences are chosen to be the central values of the global fit of Ref. [75]. We stress that, as was the case for the numerical benchmark of Table 2a, 3ν mixing angles and CPV phases obtained while identifying V with a unitary 3 × 3 mixing matrix are expected to deviate slightly from the mixing angles and CPV phases arising in a parametrisation of the full 6 × 6 mixing matrix V , due to deviations from unitarity. In are kept. 8 The horizontal green band highlights the range of sin 2 2ϑ (4) µe preferred by the global fit of Ref. [76] and cited at the beginning of this section. The dark green contour instead delimits the region inside which relatively loop-stable points can be found, i.e. points which, after the one-loop correction of Eq. (4.3) has been implemented, still have 3ν mass-squared differences and mixing angles (extracted from V ) inside the 3σ ranges of the fit [75]. From the figure it can be seen that raising the scale of M 3 will lower the scale of the light neutrino masses, disallowing too large values of m min . The approximations used in deriving the oscillation formulae of Eqs. (5.3) and (5.6) hold for all the plotted points.
Some quantities of potential phenomenological relevance, unrelated to neutrino oscillations, include the effective electron neutrino mass in β-decay, m β , the absolute value of the effective  [76], while the vertical red exclusion band is obtained by combining the most stringent Planck bound on the sum of light neutrino masses, ∑ i m i < 0.12 eV (95% CL) [4], with the 3σ ranges of mass-squared differences. The dark green contour delimits the region inside which loop-stable points have been found (see text), while the benchmark of Table 2a is marked in yellow.
neutrino Majorana mass controlling the rate of neutrinoless double beta ((ββ) 0ν -)decay, |m ββ |, and the µ → eγ branching ratio, BR(µ → eγ). For all numerical examples pertaining to case Ia which are stable under loop corrections, the latter is unobservably small BR(µ → eγ) 10 −30 , while the former two are bounded by m β < 9.4 meV and |m ββ | < 6.7 meV, and hence still out of reach of present and near-future experiments. In the computation of |m ββ |, the effects of the eV-and keV-scale neutrinos have been taken into account.
In the presence of a relatively large active-sterile mixing, future KATRIN-like experiments may be sensitive to the existence of sterile neutrinos with O(eV) masses [80]. This sensitivity is controlled by |R e1 | 2 = |V e4 | 2 , which is found to be bounded by |R e4 | 2 0.02 for the loop-

Case Ib
The numerical data for the benchmark corresponding to this case is given in Table 2b. Apart from the three light mostly-active neutrinos, the spectrum includes three mostly-sterile neutrinos with masses M 1 ∼ M 2 ∼ 3 eV, such that M 2 2 − M 2 1 1 eV 2 , while M 3 ∼ 10 14 GeV.
For the spectrum of case Ib, one has n = 2 in Eq. (3.10). In a LBL context, the expression of Eq. (3.11) applied to the transition probability of muon to electron (anti-)neutrinos can be approximated by the same expression (5.2) given for case Ia. Once again, the last two terms in that equation explicitly show the effects of the two lightest mostly-sterile states in oscillations. If one is in a condition similar to that of the benchmark of Table 2b, for which |(ΘΘ † ) µµ (ΘΘ † ) ee − 1| and |(ΘΘ † ) µe | 2 are negligible, this expression can be further approximated by: µe + sin 2 2ϑ where we have used the unitarity of the full 6 × 6 mixing matrix, and the fact that |R α1 | 2 ∼ |R α2 | 2 ≫ |R α3 | 2 . The latter prevents us from neglecting |R α2 | 2 (and hence sin 2 2ϑ (5) µe ) with respect to |R α1 | 2 (and sin 2 2ϑ (4) µe ), as we did in the previous case.
In a SBL context, the relevant form of Eq.
where terms depending on the large ∆ 4 j and ∆ 5 j ( j = 1, 2, 3) have been replaced by their averaged versions. It is clear that this case does not correspond to a typical 3+2 scenario (see for instance [76]), since one has ∆m 2 4 j , ∆m 2 5 j ∼ 10 eV 2 for j = 1, 2, 3. Hence, one can be sensitive to oscillations due to the mass-squared difference ∆m 2 54 ∼ 1 eV 2 , while oscillations pertaining to larger differences are averaged out and those driven by smaller mass-squared differences are underdeveloped. If one is in a condition similar to that of the numerical benchmark, this expression can be approximated by: µe + sin 2 2ϑ where once again we have taken into account the unitarity of the full mixing matrix and the fact that |R α1 | 2 ∼ |R α2 | 2 ≫ |R α3 | 2 . Notice that, unlike the typical 3+2 case, oscillations here depend on the square of the cosine of the relevant ∆ i j .
To further explore the parameter space of case Ib, we have produced numerical seesaw structures qualitatively similar to the benchmark by following the procedure described while discussing    Table 2b is marked in yellow.
case Ia. We have specified (at tree level) M 1 = 3.00 eV and M 2 = 3.16 eV, and have considered three different values for the heaviest neutrino mass, M 3 ∈ {10 13 , 10 14 , 5×10 14 } GeV. In Figure 2 we show the values of the average of sin 2 2ϑ (4) µe and sin 2 2ϑ 10 −30 is unobservably small, while one finds m β < 9.3 meV and |m ββ | < 4.6 meV, still out of reach of present and near-future experiments. In the computation of |m ββ |, the effects of both eV-scale neutrinos have been taken into account. One additionally finds the bounds |R e4 | 2 , |R e5 | 2 0.01 for the loop-stable numerical examples of this case.

Case II:
The numerical data for the benchmark corresponding to this case is given in Table 2c. Apart from the three light mostly-active neutrinos, the spectrum includes a mostly-sterile neutrino with mass M 1 ∼ 1 eV and a pair of quasi-degenerate neutrinos with masses M 2 M 3 ∼ 1 TeV. From Table 2c one sees that the symmetry in the last two columns of R (recall section 4.2.2) is tied to an analogous symmetry in the last two rows of X and of O c . The latter can be understood from Eqs. (2.11) and (2.14).
For the spectrum of case II, one has n = 1 in Eq. (3.10). In a LBL context, the expression of Eq. (3.11) applied to the transition probability of muon to electron (anti-)neutrinos can be approximated by: where terms depending on ∆ 4 j 1 have been replaced by their averaged versions. If one is in a condition similar to that of the benchmark of Table 2c, for which |(ΘΘ † ) µµ (ΘΘ † ) ee − 1| and |(ΘΘ † ) µe | 2 are negligible, this expression can be further approximated by: Here, we have used the unitarity of the full 6 × 6 mixing matrix, and the approximate symmetry R α2 i R α3 . If, additionally |Θ α4 | 2 = |R α1 | 2 |R α2 | 2 |R α3 | 2 , the last term can be neglected and one recovers Eq.
with ∆ 41 ∆ 42 ∆ 43 . One is thus sensitive to oscillations due to the scale of mass-squared differences ∆m 2 4 j with j = 1, 2, 3, while the oscillations pertaining to smaller mass-squared differences have not yet developed. If one is in a condition similar to that of the numerical benchmark, this expression can be approximated by: where once again the unitarity of the full mixing matrix has been taken into account, as well as the relation R α2 i R α3 . If also |R α1 | 2 |R α2 | 2 |R α3 | 2 , then the two terms containing R α2 in this equation can be neglected and one recovers Eq. (5.6) of case Ia.   CL interval of Ref. [76], and the vertical red band corresponds to the Planck constraint, as in Figure 1. The dark green contour delimits the region inside which loop-stable points have been found, while the benchmark of Table 2c is marked in yellow.
To further explore the parameter space of case II, we have produced numerical seesaw structures qualitatively similar to our benchmark by following a procedure similar to that of case Ia. We As before, the horizontal green band highlights the range of sin 2 2ϑ (4) µe preferred by the global fit of Ref. [76] and cited at the beginning of the present section, while the dark green contour delimits the region inside which relatively loop-stable points can be found. The approximations used in deriving the oscillation formulae of Eqs.  Table 2c one has BR(µ → eγ) 2.0 × 10 −13 . Such effects can be probed by the MEG II update [82], which is expected to increase the present sensitivity of MEG by one order of magnitude. One also finds the bounds m β < 15 meV, |m ββ | < 27 meV, and |R e4 | 2 0.02 for the loop-stable numerical examples of this case. While KATRIN will seek to improve the current bound on m β down to 0.2 eV, values of |m ββ | 10 −2 eV may be probed in the next generation of (ββ) 0νdecay experiments [83]. Concerning the prospect of detecting the heavy neutrino pair in future collider searches, the reader is further referred to the review [84]. If, unlike our benchmark, the heavy neutrino pair would have a mass in the 1 − 100 GeV range and were sufficiently long-lived, it might lead to displaced vertex signatures [85] and produce resolvable neutrinoantineutrino oscillations at colliders [86]. Finally, the pseudo-Dirac pair of case II might play a role in explaining the baryon asymmetry of the Universe through resonant leptogenesis [87]. In such a scenario, one should carefully take into account the washout from the interactions of the lighter sterile neutrino species. 9 These interactions may need to be non-standard in order to reconcile the light sterile neutrino paradigm with cosmology.
The presented explicit numerical examples are merely illustrative. However, they give credit to our claim that models exhibiting an approximate lepton number symmetry with at least one sterile neutrino mass at the eV scale are viable and could play a part in explaining the SBL anomalies. In the next section we look into CP Violation in the present framework in some detail.

Remarks on CP Violation Measurements
In order to analyse CP Violation effects, it is instructive to define CP asymmetries A αβ νν at the level of oscillation probabilities (see e.g. [89]): We restrict our discussion to the vacuum case, keeping in mind that in a realistic context the breaking of CP and CPT due to the asymmetry of the matter which neutrinos traverse should be taken into account. The requirement of CPT invariance results in the relations ∆P αβ = −∆P βα and ∆P αα = 0. From the unitarity of the full mixing matrix, one further has for any α, with α and β running through the whole index set, α, β = e, µ, τ, s 1 , . . ., s q . In a 3 × 3 unitary context, these relations imply that there is only one independent difference, which can be chosen as ∆P eµ . As shown in [89], in a 4 × 4 unitary framework they imply the existence of 3 independent differences, say ∆P eµ , ∆P µτ , and ∆P τe . In the 6 × 6 unitary case, we find instead that there are 10 independent differences ∆P αβ (see also [90]), while only the three of them involving just active neutrinos are experimentally relevant. Thus, one should generically expect different values for ∆P eµ , ∆P µτ , and ∆P τe in a given seesaw-type model.
Using Eq. (3.11), with n mostly-sterile neutrinos accessible at an oscillation experiment, one finds: Even if none of the new sterile states are accessible -corresponding to n = 0 -one is still expects ∆P eµ , ∆P µτ , and ∆P τe to be independent, as the relevant 3 × 3 mixing submatrix Θ (= K) is not unitary. This means that it is possible for CP invariance to hold in one oscillation channel, such as ν e and yet be violated in another, such as ν Indeed, one has:

CP-odd Weak Basis Invariants
In section 2.1 we have shown (see also Ref. [35]) that in the present framework, where three right-handed neutrinos have been added to the SM, there are 6 CPV phases. They can be made to appear in the Dirac mass matrix m by changing to the weak basis (WB) where the charged lepton mass matrix m l and the Majorana mass matrix M are diagonal and real. In the study of CP Violation, it is very useful to construct CP-odd WB invariants following the procedure introduced for the first time for the quark sector in Ref. [91], see also [92]. This procedure was later applied by different authors [35,[93][94][95][96][97][98][99] to the leptonic sector, in order to build CP-odd WB invariants relevant in several different contexts. Such invariants can be calculated in any convenient WB and their non-vanishing signals the presence of CP-breaking. We define six WB invariants which are sensitive to the leptonic CPV phases: with K L,R ≡ diag(1, e iα L,R , e iβ L,R ) and the O i j being ordinary real rotation matrices in the i-j plane, e.g. (6.12) The phases of interest are then manifestly α L,R , β L,R and δ L,R . Using this parametrisation, the invariants can be cast in the forms: ) are further sensitive to α R and β R (α L and β L ).

Summary and Conclusions
We have seen that in the framework of the type-I seesaw mechanism one can naturally have at least one sterile neutrino with a mass of around one eV. This can be inferred using a general exact parametrisation, defined in [29], that is valid irrespectively of the size and structure of the neutrino mass matrix. Thus we are able to analyse a general seesaw where not all of the three mostly-sterile neutrinos need to be very heavy. We have focused on models where at least one of the sterile neutrinos is light and its mixing with the active neutrinos is small enough to respect experimental bounds but sufficiently large to be relevant to low energy phenomenology -for instance, providing a natural explanation to the short-baseline anomalies.
In section 2, we have shown how the usual seesaw formulae have to be generalised in order to be applicable to the special region of parameters which we are considering. In particular, we have written the full neutrino mixing matrix in terms of a 3 × 3 unitary matrix and a 3 × 3 general complex matrix, which encodes the deviations from unitarity. The latter was further parametrised at tree level in terms of neutrino masses and a complex orthogonal matrix. We carefully distinguish approximate and exact relations, which are valid in any seesaw regime. Namely, we have found an exact formula for the neutrino Dirac mass matrix m in terms of neutrino masses, neutrino mixing and deviations from unitarity, which generalises the usual Casas-Ibarra parametrisation of m. We additionally derive an exact seesaw-like relation, equating the product of neutrino masses and the square of the absolute value of det m.
In section 3, we have further discussed the parametrisation of deviations from unitarity as well as constraints on said deviations in our framework. These significantly depend on the masses of the heavy neutrinos. In this context, we also find a bound on the lightest neutrino mass m min , useful whenever a light sterile is present in the seesaw spectrum. For the cases of interest, with an eV-scale sterile neutrino and large deviations from unitarity, one has m min 0.1 eV.
In sections 4 and 5 we give examples of viable textures with at least one sterile neutrino with a mass at the eV scale. Such light sterile states arise naturally by imposing an approximately conserved lepton number symmetry. Before the breaking, and for an appropriate assignment of leptonic charges, the lightest neutrinos are massless at tree level. After the breaking, the lightest neutrinos acquire calculable masses, with mass differences in agreement with experiment, after the relevant one-loop correction to the zero block of the neutrino mass matrix has been taken into account. This correction is cast in a simple form, highlighting the cancellations required by radiative stability, in section 4.1. We identify two symmetric textures (I and II) of the neutrino mass matrix which allow for a separation of high (TeV -GUT) and low ( keV) scales. We then concentrate on three particular scenarios, with differing spectra (M 1 , M 2 , M 3 ) of heavy neutrinos: case Ia, for which M 1 M 2 M 3 ; case Ib, with M 1 ∼ M 2 M 3 ; and case II, where M 1 M 2 ∼ M 3 . Numerical benchmarks are given for each of these three cases in Tables 2a -2c. Related regions in parameter space are explored in Figures 1 -3, which show that these models can accommodate enough active-sterile mixing to play a role in the explanation of short-baseline anomalies. Since the formulae for neutrino oscillation probabilities are modified in the presence of deviations from unitarity, we present, for each case, approximate expressions for muon to electron (anti-)neutrino transition probabilities, quantifying the impact of light sterile states on oscillations, for both short-and long-baseline experiments. Attention is further given to the future testability of the proposed models through non-oscillation effects of the extra neutrino states.
We conclude our work in section 6 by discussing CP Violation in the type-I seesaw framework under analysis. At the level of oscillation probability asymmetries, we have found that deviations from unitarity may source CP Violation, with generically independent effects in the standard transition channels. We have also constructed 6 CP-odd weak basis invariants which are sensitive the CP-violating phases in the lepton sector. This last point has been shown explicitly, for a particular choice of weak basis and parametrisation of m. several short term Master-type fellowships from the CFTP and CERN projects listed above. At present PMFP acknowledges support from FCT through PhD grant SFRH/BD/145399/2019. The work of JTP has been supported by the PTDC project listed above. GCB and MNR thank the CERN Theoretical Physics Department, where part of this work was done, for hospitality and partial support and also acknowledge participation in the CERN Neutrino Platform -Theory working group (CENF-TH).

A Explicit Expressions for Weak Basis Invariants
Using the definitions of section 6.2, and in the WB there considered, the WB invariants of Eq. (6.8) read: Using further the given parametrisations of V δ L,R and K L,R , one obtains the result of Eq. (6.13), with: