Dark side of the Seesaw

In an attempt to unfold (if any) a possible connection between two apparently uncorrelated sectors, namely neutrino and dark matter, we consider the type-I seesaw and a fermion singlet dark matter to start with. Our construction suggests that there exists a scalar field mediator between these two sectors whose vacuum expectation value not only generates the mass of the dark matter, but also takes part in the neutrino mass generation. While the choice of $Z_4$ symmetry allows us to establish the framework, the vacuum expectation value of the mediator field breaks $Z_4$ to a remnant $Z_2$, that is responsible to keep dark matter stable. Therefore, the observed light neutrino masses and relic abundance constraint on the dark matter, allows us to predict the heavy seesaw scale as illustrated in this paper.The methodology to connect dark matter and neutrino sector, as introduced here, is a generic one and can be applied to other possible neutrino mass generation mechanism and different dark matter candidate(s).


Introduction
The hint of physics beyond the Standard Model (SM) has come from the measurement of non zero neutrino masses and astrophysical observations supporting the existence of dark matter (DM). It is indeed intriguing to identify a common origin of both of these weakly coupled sectors.
In spite of earlier attempts to bring the dark and neutrino sector under one umbrella (see for example, [1]), one-to-one correspondence between the dark sector to a specific scenario beyond the SM responsible for seesaw mechanism to generate neutrino masses hasn't been firmly established. The main aim of this analysis is therefore to identify a simple common origin which initiates seesaw mechanism for neutrino mass generation and tames the DM phenomenology together. We point out that if the theory assumes the existence of an additional scalar singlet (φ) which couples to both DM and neutrino sector and assumes a non-zero vacuum expectation value (vev), it may generate light neutrino mass through seesaw of type-I [2][3][4][5], while also yield DM mass. Then, observed light neutrino masses and relic density constraint on DM (that crucially controls DM mass) can indicate to a particular (or a range of) value(s) of the seesaw scale, thus establishing a common origin of neutrino and dark sector. The challenge here is then to choose a symmetry that allows the interactions between dark and neutrino sector, and keep a symmetry intact to protect the DM from decaying into either neutrinos or to the SM particles after spontaneous symmetry breaking (SSB) through vev of φ. We demonstrate that the assumption of a Z 4 symmetry under which φ transforms as −1 (and suitable choices of the Z 4 charges of the other DM and SM particles) the connection can be securely established, while after SSB, the theory keeps a remnant Z 2 symmetry to stop the DM from decay. The dark sector phenomenology has been kept minimal; that is of a fermionic DM, in the form of a singlet Majorana fermion (χ) coupled to the visible sector through the singlet scalar φ. Thanks to the mixing of φ with SM Higgs due to SSB, the fermion DM can annihilate to SM particles and obtains a thermal freeze out. The seesaw mechanism is also chosen to be the simplest of its kind, type-I, assuming light neutrino mass.
The mechanism shown here is apparently the simplest of its kind, and a generic one; with many possible extensions either to relate other types of DM sector or to a different type of neutrino mass generation mechanism. The chosen set up allows a large region of parameter space where the constraints from dark sector and neutrino mass and oscillation data agrees together to indicate a limit on the Seesaw scale. One-to-one correspondence between the two sectors depend crucially how the DM mass is restricted from relic density and direct search constraints. Therefore, it has a model dependence when comes to the prediction of the Seesaw scale. The choice of the DM framework has partially been guided by the fact that the model is predictive and has a rather restrictive choices of DM mass possible from the relic and direct search data. Collider search of this particular model is difficult due to absence of charged exotic final states. One has to depend instead on irradiations of initial state particles to recoil against the DM to yield the omnipresent jets plus missing energy signature. That however, goes well with non-observation of any excess in the missing energy channels studied at Large Hadron Collider (LHC) so far.
The rest of the paper is organised as follows. We discuss the model and formalism in Section 2. Scalar potential is discussed next with the interplay of Higgs mass and related observations in Section 3. DM phenomenology comes next with relic density and direct search constraints in Section 4. In Section 5, we then discuss the allowed parameter space common to neutrino and dark matter sector to draw the connection. We finally summarise the outcome of the analysis in Section 6.

The Model
As already have been stated, we choose the simplest seesaw extension of the SM with right handed neutrino (N ), a minimal DM in the form of singlet Majorana fermion χ and a messenger singlet scalar field (φ). We consider the existence of two sectors, visible and the hidden sectors. The visible sector has the usual SM field content. However, concerning our focus on the neutrino mass generation, the visible sector simplifies to the SU (2) lepton doublets L i (i being the generation index though we omit it in the rest of our discussion for simplicity) and SM Higgs doublet H. We also consider RH neutrino to be part of this sector. On the other hand, the messenger field φ and the DM field χ together forms the hidden sector. Note that all these additional fields (i.e. beyond SM fields) transforms non-trivially under a Z 4 symmetry. A typical assignment of Z 4 charges to the fields is given in Table 1. Note that while L and N carry Lepton numbers -1 and 1 respectively, the dark sector Majorana field χ does not have any Lepton number. This Figure 1: Schematic representation for DM interaction with SM trough the scalar φ minimal field content allows us to have a phenomenologically viable DM sector and type-I seesaw mechanism for neutrino mass generation (a toy model), which are connected by the vev of the messenger field. A schematic diagram of the framework is depicted in Fig. 1 for illustration purpose. Although, SM singlet scalars charged under additional symmetries acting as portals between the dark sector and SM have been explored in some earlier attempts [6][7][8][9], this has never been connected to the neutrino sector to the best of our knowledge.

DM Sector
Field L H N φ χ Z 4 0 0 2 2 1 The Lagrangian of the framework then can be written as: where V (H, φ) is the scalar potential involving the SM Higgs doublet H and Λ is the cut-off scale of the theory. We will discuss the details of the scalar potential and its implications in the next section. Due to the Z 4 charges assigned, the only portal connecting visible and hidden sectors is the non-renormalisable 1 ΛLH φN term as well as the Higgs portal couplings in the potential as discussed in the following section. As H (and L) are neutral, the charged lepton, up and down quark sectors are compatible with the neutrino sector, with the respective charges being neutral, their Yukawa couplings arise without coupling to φ. Note that the model exibits an effective theory approach to describe the neutrino sector. Although an UV framework can be exercised, the present set-up serves as an economic one in terms of keeping the fields content and symmetry minimal.
It is important to note that here terms likeLHN ,N c N φ are disallowed due to the Z 4 symmetry imposed on the model. The other Yukawa term involves the fermion DM and the singlet scalar. The Majorana mass term for the DM fermion is disallowed by the Z 4 charges and will only be generated by the singlet scalar vev. Thus, dark matter mass M χ ∝ φ . Additionally, according to the symmetries mentioned in Table 1, a dim-5 term φ 2N c N/Λ is also allowed. However, contribution of such a term in the effective light neutrino mass is small compared to the original contribution (with M N φ 2 /Λ) and hence can be neglected. Therefore a vev of φ, will also generate light neutrino masses 6 after the seesaw mechanism suppressed by a factor of φ 2 Λ 2 , yielding a correspondence between the neutrino mass (m ν ) and DM mass (M χ = y φ ), subject to the knowledge of the seesaw scale, M N and cut-off scale Λ. The additional constraints of obtaining the correct relic abundance and direct search constraints for the DM, will control DM mass M χ to a significant extent and therefore we can estimate a limit on the heavy seesaw scale, which is the main subject of our analysis. The salient features of the model are as follows: 1. DM mass is generated through the vev of the mediator field, φ.
2. Neutrino Yukawa interaction is allowed only with a dimension-5 operator involving the same mediator field φ.
3. The above two features of the model allow us to probe the seesaw scale, M N , once the constraints from neutrino physics, DM relic density and direct detection results are incorporated. This however crucially depends on the choice of the another parameter, the cut-off scale of the theory, Λ. An interesting observation can be made with Λ = M N .
A few comments before we analyse the model under consideration. The correspondence between the dark sector and neutrino sector depends on how much one can parametrise the DM mass from relic density and direct search observation. The more the DM mass is relaxed, the less deterministic the heavy neutrino mass will be. The choice of the DM model has been motivated from above justification which we will elaborate shortly.

Scalar potential and heavy or light Higgs
The complete scalar potential, involving SU (2) L doublet H and singlet φ with the Z 4 charges as in Table 1, can be written as: as [12,13] V We note here, that the coefficient of φ 2 is deliberately chosen negative (µ 2 2 < 0) so that it acquires a non-zero vev. The term involving φ 2 H † H yields mixing between the scalars.
In the unitary gauge we can write, H = 1 , and take φ = h +u, (v, u denoting the vevs of the doublet and the singlet scalar respectively) and hence the corresponding squared mass matrix for scalar can be written as From (4), we obtain the condition for having a stable potential known as co-positivity constraints [14] and the perturbativity constraints are given by λ 1 , λ 12 < 4π. Furthermore, in such scenarios (Eq. (3)), a detailed analysis towards the vacuum stability of the potential can be obtained in [15,16]. There will be two physical Higgses (h and H ) to be originated from this theory whose mass eigenvalues are given by and their mixing is through For simplicity, we consider λ 1 = λ 2 = λ for our study and we will mostly follow this for the rest of the paper. However, given a relaxation of this constraint, we will have one more parameter to control the DM phenomenology in particular. We have added a short analysis in appendix A mentioning possible modifications when λ 1 = λ 2 and its implication in DM phenomenology.
Here we have two mass eigenstates h and H with mass eigenvalues m h and m H . Out of these two mass eigenstates, we can identify any one to be the observed Higgs boson discovered at LHC [17,18] with mass 125.7 GeV [19] depending on the choice of mixing. This can be done in two ways, namely, 1. Low mass region: Here we consider the addiitional scalar to be lighter than the SM Higgs discovered at LHC. Therefore, in this case we write these mass eigenstates as m H = m h SM and m h = m H light . Note that this is viable, as the other state is dominantly a singlet to avoid collider search bound.
2. High mass region: Here we identify the additional scalar field to be heavier than the SM Higgs discovered at LHC. In this scenario we consider the mass eigenstates to be m h = m h SM and m H = m H heavy .
It is easy to understand that the mixing has two different limits for the above two cases to be phenomenologically viable. Following our notation, sin θ ∼ 1 for the Low mass region and sin θ ∼ 0 represents the High mass scenario in the decoupling limit. We will address the two cases separately to look into their phenomenological interpretation and viable choices of the parameters to analyse the DM phenomenology in particular. Following ref [12,13], it turns out that for low mass region ( 100 GeV), the sin θ 0.9 and for the high mass region ( 150 GeV), sin θ 0.3 approximately. Without going to the very details of this sin θ dependence on the extra scalar mass and the ratio of the two vevs, we use values of sin θ within these specified range in both cases for demonstration purpose.

Low mass region
Following the notations introduced in earlier section, the relation between the mass and gauge eigenstates can be written as: Correspondingly, the masses in terms of the input parameters can be written as: To remind again, here we identify h SM to be the Higgs discovered at LHC (with m h SM = 125.7 GeV) and m H light < m h SM . Therefore, we need to choose large sin θ limit (sin θ → 1) for h SM to get dominant contribution from the SM scalar doublet H whereas H light remains dominantly a scalar singlet as evident from Eq. (7)). From the above mentioned expressions, we can recast the couplings in the scalar potential in terms of the physical quantities like masses and the mixing angels as Now with the consideration λ 1 = λ 2 = λ, we evaluate λ, λ 12 and u for fixed values of m H light and mixing angle sin θ using Eqs. (8)- (10). In Table 2    u, λ, λ 12 , for two different choices of mixing angle sin θ = 0.999 and 0.9 and three different choices of the light scalar mass m H light = 60, 80, 100 GeV. In Fig 2 we have pictorially illustrated the variation of λ and λ 12 for the low mass region. Here we observe that for a variation of the light Higgs mass m H light = 60 − 100 GeV, λ varies between 0.11 to 0.12, whereas λ 12 varies between 0.12 to 0.04 when sin θ is fixed at 0.9. Note here that there is a one-to-one correspondence between λ and λ 12 , which broadens up once we relax our assumption, i.e. λ 1 = λ 2 (see Appendix A). When λ 12 increases, λ decreases. There is another point to note here, λ 12 is also present in the triple Higgs vertex, which will control the DM phenomenology to some extent as we will demonstrate. However, we do not have a freedom of choosing it once we know both the Higgs masses and mixing. Self couplings are anyway very difficult to estimate in collider experiments. The analysis above is mainly aimed at showing the legitimacy of the input parameters for the choices of the masses of light Higgs, which we use for further analysis.

High mass region
Just like low mass case, the relation between the mass eigenstates and gauge eigenstates for the high mass region can be written as, with the corresponding physical masses given by Following the notation we established, we identify h SM as the Higgs discovered at LHC (with m h SM = 125.7 GeV) and m h SM < m H heavy . Therefore, in this alternate scenario, we work in small sin θ limit (sin θ → 0 for decoupling), where h SM is dominantly a scalar doublet and H heavy gets contribution mainly from the scalar singlet φ. This gives us the freedom to choose any mass of the other physical scalar heavier than the observed Higgs mass. To see how the physical masses are related to the input parameters, we recast the couplings as: In order to estimate the couplings in the high mass region, we choose two near end values of the mixing angle: sin θ = 0.3 and 0.001, admissible by Higgs data and find u, λ (= λ 1 = λ 2 ) and λ 12 for various values of m H heavy in Table 3. The larger is the Higgs mass and the larger is the mixing angle, the bigger will be the couplings λ and λ 12 . In Fig. 3, we have presented the correlation between λ and λ 12 for the high mass region, fixing sin θ at a moderate value of 0.1. For the heavy Higgs mass ranging between m H heavy = 200-1000 GeV, we find that λ and λ 12 varies between 0.13-0.21 and 0.025-0.26 respectively. We also see that the larger is λ, the larger will be λ 12 for this case. Again, if we relax the condition of λ 1 = λ 2 , the correspondence will be relaxed.

Dark matter relic density and direct search constraints
In this framework the dark sector is kept minimal and consists of a SM singlet fermion χ. A bare mass term for this DM is forbidden due to the specific Z 4 charge as assumed and shown in Table 1. The presence of Z 4 charged scalar field φ, yields the only Yukawa coupling yφχ c χ, involving the dark fermion as shown in the Lagrangian (Eq. (1)). The   mass of dark matter (M χ ) can only be obtained, once φ acquires vev through this interaction term. At this point, we note that there is no direct renormalisable interaction of the DM is possible with SM fields, which is true for any singlet fermion DM, unless extended to the presence of an additional doublet [20][21][22][23][24][25][26][27][28][29][30][31]. As we have already mentioned, the scalar field (φ), which connects to dark sector, also plays a crucial role in the neutrino mass generation and the explicit connection between the dark and neutrino sectors will be demonstrated shortly. Thus, prior to the spontaneous breaking of this Z 4 symmetry, M χ = 0 and the DM remains massless. Also, it doesn't have any connection to visible sector. After SSB, φ acquires a vev φ , mixes with SM Higgs doublet (H) through φ 2 H † H term and connects the DM with SM. The two phenomenologically viable scenarios discussed above after SSB, with either lower or higher mass for the other physical scalar than Higgs (the one seen at LHC), will lead to the following DM couplings through the Yukawa interactions (following earlier notation): 1. Low mass region (large sin θ):

Benchmark Points
2. High mass region (small sin θ):  as: y = M χ / φ = M χ /u. Therefore, the DM will have a s-channel annihilation through the two physical Higgses to the SM. The other possibility will be to have a t-channel annihilation to the Higgses as shown in the Feynman graphs in Fig. 4. These processes will help the fermion DM to thermally freeze out and we will compute the required values of the DM parameters to satisfy relic density constraints. The second processes (t-channel graphs) will only be feasible when the DM mass is heavier than the Higgs masses. The t-channel graphs will also play an important role to disentangle the annihilation processes to the possible direct search cross-sections that the DM obtains. With non-observation of DM in direct search experiments, such processes are essential to keep the DM model viable.
Here we again mention that the DM analysis has been performed considering λ = λ 1 = λ 2 and m h SM fixed at 125.7 GeV as observed by LHC. Subsequently, we can evaluate u, λ and λ 12 once sin θ and m H light, heavy are known. Choice of this two parameters (sin θ and m H light, heavy ) is constrained by perturbative unitarity, EW precision data, perturbativity of the couplings along with vacuum stability. We refrain from a detailed discussion in this regard, which can be found in [12,13].
Hence the DM phenomenology is completely dictated by three parameters: where M χ is the mass of the DM, sin θ is the mixing angle between two scalars and m H light, heavy is the mass of the additional scalar field. In our analysis, we use micrOMEGAs [32] to find the relic density and direct search cross sections by scanning the three-fold DM parameter space within the admissible limit.
Depending upon the mass of m H light, heavy , our analysis is categorised in two different scenarios : Low mass region (m H light < m h SM ) and High mass region (m h SM < m H heavy ) respectively to obtain constraints from DM relic density and non-observation of DM in direct search experiments.
Before we go into the details of the parameter space scan and DM constraints thereafter, a few comments are in order. The vev of φ, as is clearly seen from the Lagrangian, will break the Z 4 symmetry to a remnant Z 2 . The stability of the DM is ensured by this preserved Z 2 symmetry. The situation of a SM singlet fermion DM that is Z 2 -odd and connected to the SM through the mixing of a scalar singlet (which is even under Z 2 ) with SM Higgs has been studied in literature [33][34][35][36][37][38][39][40][41]. The model we consider here has distinct DM phenomenology, although mostly having similar features. One of the main differences is simply that the dark matter mass is proportional to the vev of the extra scalar φ (which in turn is related to sin θ). Another is the absence of the cubic φ 3 term in the scalar potential, forbidden by the Z 4 symmetry in our model, but allowed in the previously studied models where the scalar φ would be neutral. The cubic term would have altered the triple Higgs vertex, therefore changing the s-channel annihilation for the DM to the Higgs final states and also adding to the freedom of choosing the Higgs masses while keeping the input parameters within admissible range. The Z 4 symmetry in our model that is essential to connect in our model the DM to the neutrino sector, is therefore further motivated as it simplifies the DM analysis, and makes the model considerably more predictive. The scans performed in the next subsections has been systematized to the need of connecting to neutrino sector.

Dark matter phenomenology in light Higgs mass region
As has already been mentioned, in this case we assume the second neutral Higgs to be lighter m h SM > m H light with large sin θ and of course m h SM = 125.7 GeV. In Table 4 we list relevant vertices that connects the DM to visible sector and corresponding vertex factors in terms of parameters M χ , sin θ and vevs u, v and couplings λ, λ 12 with the assumption of λ 1 = λ 2 = λ. Note that the couplings are only present in the triple Higgs vertex and they do not show up elsewhere. Also we note again that the the couplings are automatically determined once we choose the light scalar mass and mixing. The vertices are introduced in the code micrOMEGAs [32] for a scan of the parameters to yield correct relic density and direct search observations.
The relic density of the DM is inversely proportional to the thermal averaged annihilation cross-section of the DM ( σv ) guided by the relation: We obtain relic density with the model implemented on the code micrOMEGAs [32]. In Fig. 5, we show the variation of DM relic density as a function of DM mass M χ . In the left, middle and right panel of Fig. 5, the BSM scalar mass m H light is fixed at 60, 80 and 100 GeV respectively. In each panel blue patch represents variation of the mixing angle sin θ in the range 0.90 ≤ sin θ ≤ 0.999 [12,13]. The region between the red horizontal lines represent correct relic density satisfying the PLANCK constraint [42]: Due to s-channel annihilation of the DM through the two Higgses (h SM and H light ), in each panel of Fig. 5, we find two resonance regions for M χ = m h SM /2 and M χ = m H light /2, where relic density drops sharply. Eventually these resonance regions intersect with the correct relic density. In these plots we also observe that, as soon as the dark matter mass becomes comparable (or more) with the mass of additional scalar H light , the dark matter annihilation is dominantly controlled by the χχ → H light H light process through t-channel graph (see Feynman diagram in Fig. 4). In Fig. 5, we also find that the relic density falls in the right ballpark when M χ ∼ m H light = 80 and 100 GeV as shown in the middle and right panel respectively. However for m H light = 60 GeV, due to large annihilation via χχ → H light H light , there is no other regions which satisfies observed relic density except the resonance regions. However the DM mass < m h SM /2 is severely constrained by the Higgs invisible data and therefore it also excludes the light Higgs resonance region.

Direct Search
Direct search of the fermion DM occurs through the t-channel graph mediated by the Higgs portal interactions as shown in Fig. 6.We again compute the direct search crosssections of the DM through the code micrOMEGAs. In Fig. 7, we show spin direct search scattering cross-section of the fermion DM in the low mass region. The plot is drawn as a function of DM mass M χ ; for three distinct choices of the BSM scalar mass: m h light = 60 GeV (left), 80 GeV (middle) and 100 GeV (right) respectively. Here also, we consider the same variation in the Higgs mixing sin θ between 0.90 to 0.999. The relic density allowed parameter space for the DM in direct search plane has also been explicitly demonstrated by the red dotted lines on the blue regions. The current experimental bounds from non-observation of DM in direct search experiments from LUX and XENON1T data are shown by dashed lines. The main outcome of this analysis is to see that the regions which satisfy both relic density and direct search constraints are those in the resonance regions, M χ = m h SM /2, m H light /2 and M χ = m H light . However keeping in mind that dark matter mass near both resonance regions is already excluded from the Higgs invisible decay, it turns out that the relic and direct search allows only M χ = m H light for this light Higgs mass region. Therefore, the model is quite restrictive in predicting the DM mass from DM constraints. This serves as a key feature to identify the connection to the neutrino sector in an unambiguous way.

Dark matter phenomenology in heavy Higgs mass region
Now let us turn to DM phenomenology of the heavy Higgs mass region, where we consider m h SM < m H heavy with small sin θ and m h SM = 125.7 GeV. First we note the vertices relevant for DM annihilations and scattering in Table 5 with appropriate vertex factors. This table is similar to Table 4 that corresponds to the low mass region excepting for the flip of notation in the triple Higgs vertices. Again, we parametrise the vertex factors in the limit of λ 1 = λ 2 = λ, which automatically get determined by the input of the heavy Higgs mass and mixing.

Relic Density
In this case relic density is plotted in Fig. 8 for five different values for m H heavy , namely 200, 400, 600 and GeV respectively. The mixing angle sin θ is varied between 0.001 to 0.3 [12,13]. Here also, in each panel, two distinct resonances can be observed at m h /2 and m H heavy /2. Apart from these resonance regions, for each choices of m H heavy , there exists a large allowed range for M χ > m H heavy , which satisfy the observed relic density by Planck data, mainly dominated by the annihilation channel χχ → H heavy H heavy . One visible difference in the high mass region from the low mass region is the large span of the mixing angle over which we can vary the parameters in high mass case. The other important point is that, the triple Higgs vertex has a proportionality to the additional Higgs mass.
With heavy Higgs, that coupling is also enhanced. Together, this yields a larger region consistent with relic density compared to the low mass case for M χ > m H heavy , which was only restricted to a small patch of M χ = m H light for low mass case.

Direct Search
The variation of the spin independent direct search scattering cross-section with the DM mass M χ is plotted in Fig. 9 for the same set of heavy Higgs masses : m H heavy = 200, 400 and 600 GeV. The blue region is obtained by scanning all the values of mixing angle in the range of 0.001 ≤ sin θ ≤ 0.3. The red points additionally satisfies relic density constraints for the choice of the specific heavy Higgs mass. Experimental direct search constraints from LUX and XENON data are shown by the dotted lines. Contrary to the low mass region, we see that a significant region of parameter space can be found below the direct search region for a wide range of DM masses. However, when we consider the relic density allowed points, we see that for a large value of DM mass for M χ >> m H heavy , we can satisfy relic density and direct search bounds in this model outside of the resonance regions. This is once again, due to the freedom in choosing a large range of sin θ and the annihilation to Higgs final states, which is not constrained by the direct search crosssections. Obviously, for larger m H heavy , the required DM mass to satisfy relic density and direct search bounds is also larger. One can also estimate the required Yukawa couplings given a specific choice of the heavy/light Higgs mass. In Fig. 10 we plot y vs M χ using the relation y = M χ /u for both low mass (Eq. 9) and high mass (Eq. 14) regions with the assumption λ = λ 1 = λ 2 . We show the cases of the same choices of the benchmark values of the heavy and light Higgs masses for both high and low mass regions, spanning the admissible range of mixing angle. We see that for larger DM mass, the required Yukawa coupling is also larger. Also for smaller BSM Higgs mass, the required Yukawa coupling is larger. Therefore, larger values of DM masses are disfavoured by the high Yukawa couplings specifically when the BSM Higgs mass is chosen smaller. The implication of this graph is then, the DM framework lives mainly in the resonance region. For low mass region, the DM constraint allowed points with M χ m H light has admissible Yukawa couplings and therefore can be considered, but for the high mass region, allowed points with M χ >> m H heavy are disfavoured by large Yukawa couplings. Here also DM mass < m h SM /2 is severely constrained by the Higgs invisible data and therefore excludes the Higgs resonance region. In table 6, we have listed some benchmark points from both low and high mass regions that are allowed by DM phenomenology.

Seesaw mechanism and the connection to dark matter
As has already been mentioned, beyond generating the dark matter mass, the scalar field φ is also instrumental for generating the Yukawa couplings of the neutrinos which then lead to the light neutrino mass through type-I seesaw. From Eq. (2), the seesaw formula in the present scenario can be written as Recent cosmological observation by Planck suggests that sum of absolute masses of three light neutrinos to be m i ≤ 0.23 eV [42]. Using Eq. (21), the bound on the masses of the light neutrinos can be written as Therefore, once we have an idea of the DM mass from the relic density and direct search constraints, as we have already obtained in the previous section, we employ Fig.  10 (left and right panels for low and high mass regions respectively) to determine the corresponding Yukawa y. The allowed regions for the right handed neutrino mass (M N ) and cut-off scale of the theory (Λ) can then be obtained from the above constraint on the combination Λ 2 M N following Eq. (22). For example, in the low mass region, from Fig.  7 we find that with m H light = 60, 80 and 100 GeV both relic density and direct search constraint can be satisfied for DM mass M χ = 29.31, 77.04 and 97.4 GeV respectively. Hence following Eq. 22, we draw a correlation between the right handed neutrino mass M N and cut-off scale Λ in Fig. 11. The shaded regions in all the panels are the allowed regions by all constraints.
Similarly, for high mass region, we draw the similar correlations between right handed   neutrino mass and the cut-off scale for different choices of Heavy Higgs mass which unambiguously point out to specific DM masses to satisfy relic density and direct search constraints as given in Fig. 12. Here we have drawn the correlations for DM masses 216, 1251 and 2815 GeV corresponding to m H heavy = 200, 400 and 600 GeV respectively. The heavier is the DM mass, the heavier will be the required right handed neutrino mass and the subsequently the cut-off scale.

Mass of the
We consider now in more detail the correlation between neutrino sector and dark matter sector. In the left panel of Fig. 13 we have plotted u(= M χ /y) vs sin θ for λ 1 = λ 2 = λ for ranges of m H light . Here the magenta, brown and dark red dots represent allowed points satisfying only dark matter relic density (obtained from Fig. 5) for m H light = 100, 80 and 60 GeV respectively. Whereas blue dots on each line also satisfies direct search limit obtained from Fig. 7. Here we find that only sin θ values close to 1 fall in the correct ball park or all the DM constraints. This plot gives an estimation for the scalar singlet vev, u, for each low mass case. In the right panel of Fig. 13 panel (following the analysis of dark matter sector in the previous section). This imposes a stringent constraint on the lower limit of the cut-off scale Λ (and RH neutrino mass). Hence for the low mass region (with m H light = 100, 80 and 60 GeV), we find the lower limit on Λ(M N ) to be Λ(M N ) ≥ 2.2 × 10 6 , 1.8 × 10 6 and 1.5 × 10 6 GeV respectively. A similar analysis can be performed for the high mass region. In the left panel of Fig. 14, we have again plotted u(= M χ /y) against sin θ for λ 1 = λ 2 = λ for various range of 200 ≤ m H heavy ≤ 400 GeV and 400 ≤ m H heavy ≤ 600 GeV as depicted by green and blue shaded regions. In this panel the magenta, brown and dark red dots represents the allowed points satisfying correct dark matter relic density only as given in Fig. 8 for m H heavy = 200, 400 and 600 GeV respectively. Blue dots additionally satisfies direct search constraints as obtained following Fig. 9. Here relatively large region of sin θ satisfies all the DM constraints representing a wide range for scalar singlet vev u. In the right panel of correct dark phenomenology. From the intercepting regions once again we can obtain the corresponding lower limit on Λ (and M N ). Here we find a relatively wider lower limit for Λ(M N ) ≥ 3.4 × 10 6 , (5.0-5.5)×10 6 and (5.7-7.1)×10 6 GeV for m H heavy = 200, 400 and 600 GeV respectively due to due larger allowed region for sin θ (and hence corresponding u in the left panel). Now if we compare the left panels of Fig. 13 and 14 we find that for low mass region, only sin θ values close to 1 (denoted by the blue dots for m H light = 100, 80 and 60 GeV respectively) satisfies direct search constraint. Hence the lighter Higgs is dominantly scalar singlet and s-channel contribution to the DM annihilation almost vanishes and t-channel diagrams dominantly contributes both in relic density and direct search constraints. In contrast, for the high mass region bound on sin θ is a bit relaxed (also denoted by the blue dots for m H heavy = 200, 400 and 600 GeV respectively) in order to satisfy both relic density and direct search constraints. This eventually leads to the fact that for low mass region, for a specific value of m H light the lower limit for the cut-off scale Λ (or RH neutrino mass M N ) is tightly constrained as evident form right panel of Fig. 13. But for high mass region due to large allowed range for sin θ, for a fixed value of m H heavy we have a wide allowed range for the lower limit for the cut-off scale Λ (or RH neutrino mass M N ). This is shown in the right panel of Fig. 14.

Summary and Conclusions
We have successfully embedded a seesaw and DM framework together, where the two sectors are related to each other by a messenger. The relation is mainly restricted by the vev of the scalar singlet field which yields both DM mass and controls the neutrino Yukawa coupling in the type-I seesaw within an effective theory framework. The first part of the framework described above constrains the DM mass from relic density and direct search constraints, given a knowledge of the additional Higgs mass and mixing. Given this information, we have explored the correlation between the right handed neutrino masses with the cut-off scale present in the theory. The main success of the set-up is to establish the correlation between DM sector and neutrino sector in a coherent manner. The scenario naturally accommodates two Higgses, one of which can be identified with the Higgs discovered at the LHC. We study both the cases where the additional Higgs field (other than the SM one) is heavier and lighter than the SM Higgs. We find that with the second Higgs as the lighter than the SM Higgs, the allowed DM phenomenology restricts DM ∼ m H light , which in turn predicts M N and Λ to be larger than 10 6 GeV. On the heavy Higgs region, this connection is little relaxed as a larger region of allowed parameter space is possible for DM. For simplicity, here we consider the quartic couplings of the two scalars present in the theory as same. The analysis can easily be extended for different values of these couplings, which would yield an extended parameter space (as hinted in the Appendix A).
As we have already stated, the choice of the dark sector was chosen as a specific exam- ple only, and one may do a similar model building exercise to connect seesaw mechanism to some other DM sector. On the other hand, the correlation requires the knowledge of the heavy or light Higgs mass and its mixing with the SM Higgs doublet, which is difficult to find at the current status of collider search experiment. As our framework involves the two heavy scales, namely the RH neutrino mass M N and the cut-off scale Λ, it would be interesting to find an UV complete construction, although this is beyond the scope of the current work and requires involvement of more fields and symmetry. (κ = 0.1) dotted lines respectively for all panels in Fig. 15 and 16. While in the low mass DM region only resonance regions satisfy DM constraints, the effect of this change (in terms of κ) is much more pronounced when the annihilation opens to the other light (or heavy) Higgs for M χ > m H light (or m H heavy ). This basically lead to a much larger allowed (by both relic density and Direct search constraints) parameter space specifically in the high DM mass region.