The seesaw portal in testable models of neutrino masses

A Standard Model extension with two Majorana neutrinos can explain the measured neutrino masses and mixings, and also account for the matter-antimatter asymmetry in a region of parameter space that could be testable in future experiments. The testability of the model relies to some extent on its minimality. In this paper we address the possibility that the model might be extended by extra generic new physics which we parametrize in terms of a low-energy effective theory. We consider the effects of the operators of the lowest dimensionality, $d=5$, and evaluate the upper bounds on the coefficients so that the predictions of the minimal model are robust. One of the operators gives a new production mechanism for the heavy neutrinos at LHC via higgs decays. The higgs can decay to a pair of such neutrinos that, being long-lived, leave a powerful signal of two displaced vertices. We estimate the LHC reach to this process.


Introduction
The measured neutrino masses and mixings provide a strong hint of physics beyond the Standard Model, and this new physics might involve new weakly coupled fields at the electroweak scale. The simplest of these possibilities is an extension of the Standard Model with just two singlet Majorana fermions. This model has been shown to provide a natural explanation of the matter-antimatter asymmetry of the Universe [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15] via neutrino oscillations [16] if the two heavy singlets have masses in the range [1, 10 2 ] GeV, a range that implies that these states could be produced and searched for in beam dump experiments (for a recent review see [17]) and colliders (for a sample set of references see [18][19][20][21][22][23][24][25][26][27][28][29]). This opens the possibility of making this baryogenesis scenario testable: the combination of measurements of the mixings and masses of the heavy neutrino states, together with the determination of the CP phase δ in future oscillation experiments and the amplitude of neutrinoless double beta decay might lead to a quantitative prediction of the baryon asymmetry in the universe [13].
The constraint of the measured neutrino masses and mixings fixes to a large extent the flavour mixings of the heavy neutrino states [13]. In particular, it has been shown that the ratios of mixings to different lepton flavours are strongly correlated with the CP violating phases of the PMNS matrix [30]. One of the most drastic implications of this minimal model is the vanishing of the lightest neutrino mass. If evidence of a nonvanishing lightest neutrino mass would come from future beta-decay experiments or from cosmological measurements, the minimal model with two neutrinos would not be able to explain it and an extension would be required.
The question we would like to address in this paper is the following. If there is some extra new physics at a higher scale (for example more singlet states), how does this new physics would modify the predictions of the low-energy theory represented by the minimal model with the two singlet states. In particular, we would like to understand if the strong correlations between the light and heavy neutrino masses and mixings [30] that underlie the predictivity and testability of this model could be preserved or under what conditions they might be.
A model independent way of answering this question is by building the effective theory and analysing what modifications on the correlations higher dimensional operators can induce. The list of higher dimensional operators in the SM up to dimension d = 6 is wellknown [31][32][33] and has been studied extensively. Interestingly however in the extended theory with fermion singlets [34][35][36], there are more d = 5 operators than in the SM. The relatively light singlet states provide a new portal into BSM physics. In this paper we will restrict ourselves to the lowest dimensional operators of d = 5, which are expected to be dominant.
The paper is organized as follows. In the section 2 we describe the minimal model and the extension by d = 5 operators. In section 3 we consider the different constraints on the d = 5 operators from neutrino masses and LHC and in section 4 we conclude.

The seesaw effective theory
At energies much smaller than the new physics scale, Λ, the theory is just a type I seesaw model [37][38][39][40] with two Majorana neutrinos in the GeV range, with the Lagrangian The leading effects of the new physics should be well described by higher dimensional operators of d = 5 that can be constructed in a gauge invariant way with the Standard Model fields and the heavy Majorana neutrinos. These have been classified in Refs. [34][35][36]. There are three independent operators: 3) The first is the well known Weinberg operator O W [31] that induces a new contribution to the light neutrino masses, independent of the contribution of the N fields. The new operator O N Φ contributes to the N Majorana masses, and interestingly gives additional couplings of these heavy neutrinos to the Higgs [19,34], which are not necessarily suppressed with the Yukawa couplings. Constrains on this operator have been extensively studied in the context of Higgs portal dark matter [41]. In that case however, it is assumed that the Majorana fermion constitutes the dark matter and therefore does not decay, for which it is necessary to forbid the yukawa coupling to the lepton doublet. In our case the states can decay visibly in the detector. The last operator induces magnetic moments of the heavy neutrinos and the constraints have been studied in [36].
We will see in the following that the direct constraints on the coefficients of these operators are very different. It is therefore an important question whether large hierarchies could exist between the coefficients. On naturality grounds we would expect they should of the same order However this might not be the case if there exists an approximate global U (1) L symmetry [42,43]. If the two Majorana neutrinos carry opposite lepton number charges, the Majorana mass term can be chosen to be of the form to ensure an exact symmetry, implying degenerate heavy neutrinos and massless light neutrinos. We note that this approximate U (1) L symmetry is also usually invoked in the minimal model to justify yukawa couplings larger than the naive seesaw relation Y ∼ Only the Weinberg operator violates lepton number in this case. If we assume the symmetry is approximately conserved, since we need to accommodate neutrino masses, it is technically natural in this context to assume the following hierarchy A hierarchy also arises in the context of minimal flavour violation [44][45][46]. As discussed in [34], in this case the coefficients On the other hand, if we drop completely the naturalness argument, we have to confront a hierarchy problem. All the d = 5 operators generically mix under renormalization. For example the diagram of Fig. 1 induces a contribution to the Weinberg operator at one loop of the form Requiring this contribution to be of the same order or smaller than the tree level contribution implies the following bound where θ is the mixing of the heavy states with the leptons. This constraint is rather mild and leaves significant freedom to have a large hierarchy, even if no symmetry principle is assumed.
3 Constraints on the d = 5 operators

Neutrino masses
The operator O N Φ gives a correction to the Majorana neutrino mass matrix: The light neutrino masses get a contribution from the two light states, and from new physics at the scale Λ. We call this latter contribution δm W ν : The diagonalization of the complete light neutrino mass matrix gives: where U is the standard PMNS matrix, m l = diag (m 1 , m 2 , m 3 ) are the light neutrino masses. We know that in the limit δm W ν → 0 , m 1 (m 3 ) → 0 for the normal (inverted) hierarchy (NH/IH) or equivalently the condition must hold. Perturbing in (δm W ν ) ij = δ ij and the small light neutrino parameters θ 13 ∼ θ 23 − π/4 ∼ r ≡ ∆m 2 sol /∆m 2 atm , all of which are taken of the same order, O( ) ∼ O(δ), we obtain at leading order: Barring fine-tuned cancellations among the δ ij coefficients, we have therefore The lightest neutrino mass is thus the measure of the non-standard contributions from O W .

Heavy neutrino Mixing
The heavy states mix with the charged leptons inducing charged and neutral currents as well as higgs interactions. All these get modified by the d = 5 operators. In the mass basis the heavy neutrinos interact with the Z and W via the interactions: where N i are now the mass eigenstates, i = 1, 2 and U α3+i are their mixings to the flavours α = e, µ, τ . These mixings are related to the Yukawa couplings as We can use the Casas-Ibarra trick [47] to parametrize Y in terms of m and U where and a generic 2×2 complex orthogonal matrix R. In this parametrization the mixings are In the limit δm W ν → 0, m and U coincide with the light neutrino masses and mixing matrix. It has been shown that in this case the flavour structure of the mixings is approximately fixed those neutrino oscillation observables [13,46,48]. Simple analytical approximations were worked out in [13] that should be accurate in the sensitivity region of future experiments. These are shown in Fig. 2, where we show the allowed range for normalized flavour mixings: |U α3+i | 2 /Ū 2 , withŪ 2 ≡ α |U α3+i | 2 , fixing the known oscillation parameters to their best fit values [49], and varying the CP violating phases δ and φ 1 in the whole physical range (the second Majorana phase is unphysical for two heavy neutrinos). It has been shown that the determination of these ratios in future experiments would give very valuable information on the CP phases [30].
In general however the parametersŨ andm are sensitive to new physics. We can relate them to the physical mixings and masses using perturbation theory in δm W ν = O(δ).
where {, } is the anticommutator. For IH the result is the same with m 3 → m 1 and moved to the first position. Here we have used that m 1 /m 3 = O(δ). The first order corrections O (δ) tom andŨ are presented in appendix A. Including these in eq. (3.12) we find that the approximate expressions for the mixings in the absence of extra dynamics derived in ref. [13] get modified by corrections δ|U αi | 2 . For NH we get where A = e 2γ ∆m 2 atm /4. Similarly for IH: and δ|U α4 | 2 M 1 = δ|U α5 | 2 M 2 at this order for both hierarchies. Barring fine tuning among the different entries in δm W ν and taking into account Eq. are expected to be O (1). In Fig. 3 we show the impact of these corrections to the ratio |U e4 | 2 /|U µ4 | 2 dependence on the CP phases for m 1,3 = 0.1 ∆m 2 sol and varying c α and c α between −1 and 1 to maximize the difference. The corrections to the flavour ratios are therefore of O(m lightest / ∆m 2 atm ) for IH and of O(m lightest / ∆m 2 solar ) for NH. It will be hard experimentally to ensure that these corrections are sufficiently small, in other words to prove that m lightest is significantly smaller than ∆m 2 atm for IH and ∆m 2 solar for NH. On the other hand, measuring a lightest neutrino mass above this value will surely imply that corrections to the predictions of the minimal model are likely to be significant. Alternatively finding the flavour ratios in the regions given by Fig. 2 would be a strong indication that only two Majorana states give the dominant contribution to the light neutrino masses and therefore the lightest neutrino mass is significantly smaller than the mass splittings.

Higgs-Neutrino Interactions
Interestingly the operator O N Φ induces new higgs interactions with the right handed neutrinos. In particular for M i ≤ M H 2 , the Higgs can decay to two heavy neutrinos via this coupling This decay leads to a spectacular signal at LHC, which is that of a pair of displaced vertices (DVs) since the N decay only via mixing and they have a long decay length. The process is depicted in Fig. 4 Note that compared to the minimal model, where the sterile neutrinos are singly produced via mixing, in this case the production is controlled by the coefficient α N Φ and therefore one should have access to much smaller mixings, as long as they lead to lifetimes are within reach of DV searches within the detectors. Figure 4. Higgs decay to two heavy neutrinos leading to displaced vertices Our aim in this section is to do a simple estimate of the bounds on the coupling α N Φ /Λ from searches of higgs decays to two displaced vertices at LHC. A closely related calculation has been done in the context of U (1) models in [50], where the signal selection has been performed following recent searches by the CMS collaboration [51,52]. We have considered two different analyses: 1) a search of displaced tracks in the inner tracker where at least one displaced lepton, e or µ, is reconstructed from each vertex; 2) a search for displaced tracks in the muon chambers and outside the inner tracker where at least one µ is reconstructed from each vertex. The charges are not restricted and therefore events with same-sign or opposite sign leptons are possible.
For simplicity we will consider only semileptonic decays of the N i which give rise to two lepton final states through the decay (3.17) We consider a parton-level Monte Carlo analysis using Madgraph5 [53] at LHC with a center-of-mass energy of 13TeV and 300 fb −1 luminosity. We include only the dominant gluon fusion higgs production and we consider the production of just one neutrino species, N 1 . The production cross section pp → h → N 1 N 1 is shown in Fig. 5 as a function of the heavy neutrino mass for various values of the coupling g N Φ ≡ v(α N Φ ) 11 √ 2Λ . In Fig. 6 we show the Br(H → N 1 N 1 ) as a function of g N Φ for various values of the mass (here we assume the higgs decays just to one neutrino).
The p T of the two leading leptons is shown in Fig. (7). Following [50], the signal selection is done by requiring two lepton tracks, e or µ that satisfy the following kinematical cuts on transverse momentum, pseudorapidity and isolation of the two tracks: p T (l) > 26 GeV, |η| < 2, ∆R > 0.2, cos θ µµ > −0.75. (3.18) In the case of muons a constraint in the opening angle θ µµ is imposed in order to reduce the cosmic muon background. The efficiencies resulting from these consecutive cuts for  For each event the decay length of each neutrino in the laboratory frame is obtained from their simulated momenta and the distance travelled, L, is randomly sampled according  to the expected exponential distribution. The impact parameter of the lepton in the transverse direction d 0 is defined as where L x,y,z are the X, Y, Z projections of the vector L u, where u is the velocity of the heavy neutrino, and (p x , p y ) is the momentum of the lepton in the transverse direction. The cuts associated to the displaced tracks are different for the two analyses [50]: • Inner tracker (IT): (3.20) where σ t d 20µm is the resolution in the tracker.
• Muon chambers (MC): (3.21) where the impact parameter resolution in the chambers is σ µ d ∼ 2cm. Since L −1 ∝ U 2 M 6 , we expect that these cuts depend mostly on this combination of mass and mixing with some dispersion. In Figs. 8 we show the acceptance given by cuts of eqs. (3.20) and (3.21) as a function of U 2 M 6 . As expected the two analyses are sensitive to different decay lengths. We have checked that the global efficiencies resulting for each of these cuts agree reasonably well with the results quoted in [50].
Concerning the flavour structures, we have considered two scenarios: • Scenario A: only the decay of the higgs to one neutrino species, N 1 , is considered. Obviously if the other state has a mass below M H /2, it will also contribute via the decays H → N 2 N 2 and H → N 1 N 2 . These will be sensitive however to different entries of the matrix α N Φ .
In Fig. 9 we show the BR(H → N 1 N 1 ) corresponding to 4 signal events as a function of M H 2M × cτ which is approximately the decay length in the laboratory frame. We present the results that include the events producing two displaced electrons in the inner tracker, and either two displaced µ in the inner tracker (IT analysis) or two displaced muons in the muon chamber (MC analysis). We compare these limits with the those of the Mathusla project of the HL-LHC with 3000f b −1 as obtained in ref. [54], where all the decays of the N 1 are assumed observed. The limit this implies on the coefficient of the operator, g N Φ can then be read from Fig. 6.
As we have seen the flavour structure of the mixings depends strongly on the hierarchy and the CP phases (see Fig. 2). We fix the CP phases to δ = 0, φ 1 = π/2. For this choice and NH, the muon channels dominates, while for IH it is the electron that dominates. In the former case therefore the reach of the two analysis IT and MC are quite different, showing the reach in complementary regions of cτ , while for the latter most of the sensitivity comes from the electron search which contributes the same to both analyses as it can only be done with the tracker. The dependence on the phases is illustrated in Fig. 10 where the phase φ 1 is chosen approximately to maximize/minimize the sensitivity. This pattern ensures that a lepton number is preserved under which N 1 and N 2 have opposite charges. In the same basis only N 1 has Yukawa couplings. As a result of the approximate symmetry same sign lepton decays vanish, the massive state is a Dirac fermion. The branching ratios are however unchanged. The bounds on this scenario also can be read from Fig. 9, if we interpret the y-axis as the total branching ratio of the higgs to the heavy Dirac state. On other hand the dependence of this branching ratio on the coupling g N Φ is twice larger as that shown in Fig. 6.

Discussion
In order to explain the observed neutrino masses in the context of Type I seesaw models, only two heavy Majorana singlets are required. This is the minimal extension of the Standard Model that can accommodate neutrino masses. The flavour structure of the mixings of the heavy neutrino mass eigenstates is strongly correlated with the light neutrino masses and the PMNS matrix. If the mass of these heavy states is in the electroweak Figure 10. Expected limits at LHC (13 TeV, 300 fb −1 ) on the BR(H → N 1 N 1 ) for scenario A as a function of M H 2M cτ in meters. The mass is 20 GeV and the CP phase φ 1 = −π/2 (blue) or φ 1 = π/2 (red) for NH (solid) and IH (dashed). range, they can be searched for in future experiments within the high intensity frontier. A putative measurement of their masses and mixings might demonstrate the origin of neutrino masses. The predictivity of this minimal model relies on the assumption that only these two states give the dominant contribution to neutrino masses. Any extension of this minimal scenario can modify these predictions. We have considered the possible modifications on the minimal scenario induced by generic new physics at some higher scale Λ. Three operators can parametrize these modifications at leading order in Λ −1 .
Assuming the coefficients of these operators are all of the same order, the strongest bound comes from the bound on the lightest neutrino mass: for NH, and this is only warrantied in the hierarchical scenario where m light is smaller than 0.05eV. A possible extension of the minimal scenario that gives rise to the Weinberg operator is obviously the addition of extra Majorana singlets. In particular, in the case where one more neutrino is added, it is necessary that the mass is larger and/or the mixings smaller. This is for example the case in the νMSM [1], where the extra keV state is very weakly coupled, so that the contribution to the lightest neutrino mass is below 10 −5 eV, well below the above requirement. On the other hand, large hierarchies α W α N Φ ∼ α B could be present undisrupted by radiative corrections. In this case, direct bounds on the other two d = 5 operators might be competitive and offer a new window into neutrino physics at the LHC. We have considered the bounds on α N Φ from searches of displaced leptons at LHC and we have found that LHC with 300 fb −1 at 13TeV could set bounds It is important to note that if the coefficient of this operator is above this sensitivity limit, LHC could detect the sterile neutrinos for significantly smaller mixings than it is possible in the minimal model, in particular LHC could even reach the seesaw limit as shown in Fig. 11. The bounds on α N B were considered in ref. [36] and found to be α N B Λ ≤ 10 −2 − 10 −1 TeV −1 . Finally flavour symmetries could also explain a large hierarchy of α W and α N Φ or α N B . An approximate U (1) L could explain the hierarchy α W α N Φ , α N B . Such symmetry is also the most natural scenario within the minimal model to have mixings significantly larger than the naive seesaw limit as required for their observability. Similarly, hierarchies of this type or α W , α N B α N Φ are expected in the context of minimal flavour violation [34].
The presence of large contributions from these operators are not expected to modify the predictions concerning the flavour structure of the mixings of the heavy neutrinos, since the higher order corrections to the neutrino masses are very small, but they could provide a new portal at colliders to reveal the mechanism behind neutrino masses. On the other hand the same interactions will surely affect the baryogenesis scenario with respect to the minimal model, since the sterile neutrinos can reach thermal equilibrium if the interactions induced by the higher dimensional operators are within LHC reach. The impact of O N Φ in the context of baryogenesis has been recenlty considered in [55].