Right sneutrino with $\Delta\,L\,=\,2$ masses as non-thermal dark matter

We consider MSSM with right-chiral neutrino superfields with Majorana masses, where the lightest right-handed sneutrino dominated scalars constitutes non-thermal dark matter (DM). The $\Delta\,L=2$ masses are subject to severe constraints coming from freeze-in relic density of such DM candidates as well as from sterile neutrino $\textit{freeze-in}$. In addition, big-bang Nucleosynthesis and $\textit{freeze-out}$ of the next-to-lightest superparticle shrink the viable parameter space of such a scenario. We examine various $\Delta\,L=2$ mass terms for families other than that $\Delta\,L=2$ masses are difficult to reconcile with a right-sneutrino DM, unless there is either (a) a hierarchy of about 3 orders of magnitudes among various supersymmetry-breaking mass parameters, or, (b) strong cancellation between the higgsino mass and the trilinear supersymmetry breaking mass parameter for sneutrinos.


Introduction
While the search for the dark matter (DM) candidate(s) of our universe is on, one constantly feels the necessity of going beyond stereotypes in modelling physics beyond the standard model (SM) to accommodate the candidate particle(s). It is in this spirit that alternative candidates in small extensions of the minimal supersymmetric standard model (MSSM) have been explored. Scenarios with gravitino as warm DM are thus frequently discussed [1][2][3][4][5][6][7][8][9][10]. Another, quite minimalistic, extension is to extend the MSSM with a right-chiral neutrino superfield for each family, and postulate that one right-sneutrino-dominated scalar is the DM candidate. Various cosmological as well as phenomenological implications of this scenario have already been explored [10][11][12][13][14][15][16][17][18].
The interaction of such a particle with all other MSSM fields is proportional to the very small neutrino Yukawa coupling. Consequently, this scenario almost always leads to a non-thermal DM, with a long-lived next-to-lightest SUSY particle (NLSP). Such a longlived NLSP mostly survives till it decouples from the thermal bath, and decays to the DM candidate thereafter. Usually, such scenarios are explored by assuming that neutrinos have just Dirac masses, in which cases their Yukawa coupling strengths are O(10 −13 ) [14,[19][20][21][22][23][24][25][26].
Some additional effects are operative if neutrinos have Majorana masses as well. This will require the right-handed neutrino fields to have ∆ L = 2 mass terms. The Type-I seesaw mechanism works here, thus requiring somewhat higher neutrino Yukawa couplings. This, however, entails the process of freeze-in of right sneutrinos while the NLSP (a stau, for example) is yet to decouple. Studies including those on similar non-SUSY theories have shown that such freeze-in contribution to the relic density impose rather strong constraints on the corresponding model parameters [27][28][29][30]. Since the freeze-in rates are proportional to the square of the Yukawa coupling(s), this effect is insignificant for Dirac neutrino scenarios [14,[24][25][26]. In a case with ∆L = 2 mass terms present, however, the seesaw formula allows bigger Yukawa couplings, and thus the constraints tighten [31][32][33][34][35]. One has to also fit the neutrino mass and mixing patterns [36], and, quite seriously, the potential contribution of long-lived sterile neutrinos generated via the Dodelson-Widrow mechanism [37][38][39][40][41][42][43][44]. Thus the ∆L = 2 Majorana masses, not only corresponding to the family where the right sneutrino DM candidate belongs but also for the other families, get seriously restricted, especially in view of the PLANCK data [45]. The way such restrictions arise is investigated in the present paper.
The right-handed Majorana mass matrix is assumed to be diagonal without loss of generality. Since the present neutrino oscillation experiments do not constrain mass of the lightest active neutrino we can constrain only two of the three heavy Majorana mass terms in the Majorana mass matrix.
In the pure Dirac case, the relic density of the non-thermal DM is obtained from post-freeze-out decay of the NLSP [14]. When freeze-in has a role to play, significant contribution to relic density also occurs in the decay of heavier superparticles still in thermal equilibrium [31]. The decay to the DM is governed essentially by the neutrino Yukawa coupling, even when it is driven by gauge interactions. Since such interactions will affect the (small but non-vanishing) left-sneutrino component in the DM, the deciding factor is the sneutrino mixing angle, which essentially depends on the off-diagonal part of the sneutrino mass matrix. This block is proportional to the Yukawa coupling in usual parametrisations. Of course, the mixing is also reduced if one has the SUSY-breaking terms in the diagonal block are much larger than those in the off-diagonal block. This, however, requires a hierarchy of SUSY-breaking mass parameters, which is prima facie inexplicable, unlesss a cancellation between F-terms and soft terms in the off-diagonal block is engineered.
There can be several scales associated with the ∆L = 2 masses, consistent with the Type-I seesaw mechanism. The commonly known GUT-scale seesaw won't work for a case where non-thermal DM candidates are sought, as that would entail Yukawa couplings large enough for them to thermalise. Majorana masses in the electroweak scale [46][47][48], too, imply Yukawa couplings that could enhance the freeze-in rate unacceptably, unless a suppression in left-right mixing occurs via diagonal SUSY breaking terms that are several orders of magnitude larger than the off-diagonal ones. Though the situation is marginally better for O(1) GeV Majorana masses, a fine-tuned cancellation between the soft-and Fterms is required there, too. For hierarchical neutrino masses, the lightest neutrino mass being a free parameter, one can not easily constrain the corresponding ∆ L = 2 mass. We have considered this mass to be at the keV scale in our analysis, which does not hamper the generality of the constraints on the other Majorana masses derived here.
The scenario with all the three Majorana masses ranging from 500 MeV down to a few tens of keV is constrained by light element abundance. When all three masses are in keV scale, they all become warm dark matter, a scenario ruled out already due to overproduction of the warm keV scale dark matter via Dodelson-Widrow mechanism. We may have one ∆ L = 2 mass in the eV scale [49], as considered in [34], but such a situation brings in severe cosmological constraints [45,50].
We establish freeze-in of right-sneutrino DM to be a major constraining factor with ∆ L = 2 masses. In such a case, not only the NLSP but also the rest of the MSSM spectrum contributes to freeze-in. This constraint is thus largely applicable to scenarios including various NLSP candidates. We focus on Stau and neutralino NLSP in particular.
Our paper have been organized as follows: In section 2 we discuss the model considered by us. Section 3 has been devoted to the discussions of constraints coming from freeze-in, NLSP freeze-out, low mass sterile neutrinos and Big-bang Nucleosynthesis (BBN). In section 4 we have shown how the constraints mentioned already can severly restrict the parameter space of such a scenario. In section 5 we have discussed the related issues of quasi-degenerate neutrino masses. We summarise and conclude in section 6. Finally the formula used by us have been tabulated in Appendices A and B.

Model
We consider the MSSM scenario augmented with three right chiral neutrino superfields (N R ), which possess ∆ L = 2 mass terms. Hence the superpotential gets extended to the form [51][52][53] whereĤ u is the higgs doublet that couples to up-type quarks,L is left-chiral SU(2) doublet lepton superfields andN R is a right-handed neutrino superfield. While Y ν is the neutrino Yukawa coupling matrix, M N is the ∆ L = 2 mass matrix for the heavy neutrinos, assumed, as already stated to be diagonal, since basis rotations in the right-handed neutrino sector is unlikely to affect our main conclusions. Since we are not assuming any high scale mechanism of SUSY breaking we need to add all allowed SUSY-breaking terms phenomenologically. The relevant soft terms in this case are where m N (m B ) corresponds to ∆ L = 0(2) susy-breaking sneutrino masses and T ν is the coefficient of the well-known trilinear SUSY-breaking term. Left-right mixing in the sneutrino sector occurs, via the soft terms proportional to T ν , and the F-terms proportional to the higgsino mass parameter(µ), when h u (h d ) acquires the vaccum expectation values v u (v d ). In addition to m N and m B , the F-term masses proportional to M N adds to the right-handed sneutrino masses. In our case, one requires M N not to exceed this SUSYbreaking scale, so that a right sneutrino may behave as a DM candidate.

Neutrino masses and mixings
An important constraint on the right sneutrino DM arises from the bearing it has on the mass and mixing patterns of light neutrinos. In this situation the simultaneous occurrence of a Dirac mass matrix (m D ) and a ∆ L = 2 mass matrix (M N ) leads to the standard Type-I seesaw expression for the light neutrino mass matrix: where c ij = cos θ ij , s ij = sin θ ij and δ is the Dirac CP phase.
To study the prospect of right-handed sneutrino being a non-thermal dark matter we need to appropriately fix Y ν . This in turn governs the decay of heavier sparticles to the right-handed sneutrino DM, as will be discussed in section 3. Combining equations 2.3 into equations 2.4, Multiplying with 1 m diag ν from left and right of equation 2.6 one obtains (2.7) Using the Casas-Ibarra parametrization [54] one can introduce a complex orthogonal matrix R as, It is clear from equation 2.9 that R thus leads us to a Yukawa matrix that satisfies the neutrino oscillation data by construction. 1 We have illustrated our main points for a normal hierarchy(NH) of neutrino masses. To compute the Yukawa coupling matrix we need the U P M N S and hence we have restricted ourselves to the central value for the neutrino oscillation parameters θ 12 , θ 23 , θ 13 as given by [36] and δ is chosen to be 3π 2 . Since the variation of Majorana phases does not change the order of magnitude of the Yukawa couplings and our conclusion is not very sensitive to such variation, we set α 1 , α 2 to be π 4 . We have chosen the lightest neutrino mass m 1 to be negligibly small (O(∼ 10 −8 eV)) while the masses of the other neutrinos are determined by the solar( m 2 ) and atmospheric( m 2 ⊕ ) mass-squared differences. The lightest neutrino being effectively massless, masses of the relatively heavier neutrinos are given by m 2 = m 2 = 0.0086 eV and m 3 = m 2 ⊕ = 0.05 eV respectively. Since one light neutrino is effectively massless, we have parametrized our R matrix as follows [55]: where a non-vanishing R 11 with appropriate m diag ν reflects the postulated hierarchy. Our analysis uses ω = i.

Sneutrino masses and mixings
Turning now to the supersymmetric sector of the concerned model, the sneutrino mass matrix in the usual flavour basis (ν L ,Ñ R ) is [34,35], are the 3×3 mass matrices in the flavour basis. ml L is the soft-mass term for charged lepton doubletL. T ν will be parametrised as T ν = A ν Y ν , A ν being a mass-scale related to the scale of SUSY-breaking.
To study the sneutrino mass terms and phenomenology of sneutrino mass eigenstates originated from equation 2.10 it is convenient to introduce the real fields,(ν i 1 ,Ñ i 1 ,ν i 2 ,Ñ i 2 ) defined as follows [34,35,53]: In the basis of these CP-even(ν 1 ,Ñ 1 ) and CP-odd (ν 2 ,Ñ 2 ) real sneutrino fields the sneutrino mass matrix is The matrix in equation 2.12 can be diagonalized by an unitary rotation through an angle θν given by, where the top sign corresponds to mixing in the CP-even sector and bottom sign to CP-odd sector respectively. It is clear from the structure of the mixing matrix in (2.12) that the CP-even sneutrinos mix among themselves, as do the CP-odd ones. The decay of NLSP to LSP is proportional to θ 2 ν and hence to Y 2 ν . The BBN constraints suggests the NLSP lifetime τ N LSP 100 sec. there by posing a strong lower-limit on Y ν . While the upper-limit on the Y ν has been set by correct relic density requirements.
Since we are considering the lightest sneutrino to be the LSP and also dominantly right-handed, its mass will be ∼ m 2 RR − m 2 B and it will be CP-even in nature. Concomitantly, the lightest CP-odd sneutrino which is dominantly right-handed will have mass ∼ m 2 RR + m 2 B and will also be a dark matter in addition to the CP-even LSP 2 . We have considered these lightest CP-even and CP-odd sneutrinos belong to the family which corresponds to the lightest active neutrino while the dominantly right-handed sneutrinos belonging to the other two families are heavy enough to decay before BBN. This assumption simplifies the calculation of relic density for the non-thermal sneutrino dark matter particle(s) which will be denoted asν DM collectively from now on.

Sources of constraints
While exploring the prospects of right-handed sneutrino being a non-thermal DM one has to take into account several constraints.
• The Yukawa interaction strength is bounded above by the consideration of freeze-in of sneutrinos.
• Post freeze-out decay of NLSP enhances the DM abundance and limits the parameter space following the observed density of DM.
• keV-MeV scale Majorana neutrino masses and the corresponding active-sterile mixings are constrained following BBN considerations, X-ray and Gamma-ray observations as well as the PLANCK data.
• Furthermore late decay of NLSP can mess up the observed light element abundances, thus implying a minimum value of the Yukawa couplings.
We discuss below these constraints in some detail.

Freeze-In of RH sneutrinos
A non-thermal DM candidate never reaches thermal equilibrium with the thermal plasma because of it's low interaction strength. It is however, produced in the decays of the MSSM particles, mostly the NLSP, till the latter, being quasi-stable, freezes-out of the thermal bath. This is called the freeze-in process for the DM particles [31,[56][57][58][59][60][61]. The relic density of a non-thermal DM candidate generated via this process is proportional to the decay width of heavier superparticles into theν DM . Since decay width is proportional to the square of the Yukawa coupling Y ν , the observed relic density as reported by the PLANCK collaboration [45] sets upper limits on Y ν and M N . The value Ω CDM h 2 = 0.1199 ± 0.0027 has been used to obtain such limits. In general, there exist a long-lived particle χ, which is decoupled from the thermal bath. Despite the lack of equilibrium, χ can be produced from the decay of heavier bath particles, albeit via feeble interactions. Such production occurs mostly when the temperature T drops below the mass of the decaying bath particles. For the sake of illustration, if a heavy particle A in the thermal bath decays to lighter SM particle B in the bath and χ then the Boltzmann equation governing the evolution of χ is given by, where n χ is the comoving number density of the particle χ, commonly known as feebly interacting massive particle (FIMP) and H is the Hubble constant. Here m A and E A are the mass and energy of the decaying particle and Γ A is the decay width of the process A → B + χ. By solving equation 3.1 for the yield of χ defined as, Y χ = n χ s where s is the entropy density of the universe, one gets If there are several heavy particle in the bath the decay of which may lead to the production of χ then one will have where x = m A T and g s,ρ * are the number of degrees of freedom at the time of freeze-in i.e. T m A for entropy s and energy density ρ respectively. In our specific context, the right-chiral sneutrino DM (ν DM ) can be produced from the decays of higgsino dominated or gaugino dominated neutralinos(χ 0 i ), charginos(χ ± i ) as well as left-chiral sleptons(l L ). The corresponding decay widths and amplitudes are tabulated in Appendices A and B respectively.
Let g s * (g ρ * ) be the number of degrees of freedom contributing to total entropy(energy) density. With M P l = 10 19 GeV, g s * g ρ * 100 [62] in equation 3.3, the freeze-in relic density ofν DM , denoted by Ω F I is obtained using the appropriate degree of freedom g A for each of the decaying particles mentioned above, Mν DM is the mass ofν DM and m susy is the generic value of the SUSY-breaking masses.
Freeze-in intoν DM depends on the decay rates of charginos and neutralinos as well as sleptons into it. Considering neutralinos, one finds that any higgsino component in χ 0 i (i=1,..4) has a decay amplitude proportional to Y ν . If we further assume that the right sneutrino of the same family as that of the lightest active neutrino is the LSP and it has negligible mixing with other flavours, then the higgsino decay width to the LSP sneutrino is solely determined by the lightest neutrino Yukawa coupling (Y ν ) 11 . We have already seen that the lightest active neutrino mass is a free parameter in the normal hierarchical scenario. 3 Therefore one can always have such a mass for it, when the contribution of any higgsino-dominated neutralino to sneutrino abundance is negligibly small compared to the observed DM abundance [45]. This choice considerably simplifies our analysis. The more general mixing scenario multiplies the number of free parameters but does not affect the constraint on the Majorana mass of the lightest neutrino. Throughout our analysis the Yukawa coupling corresponding to lightest active neutrino is O(10 −13 ), which ensures the contribution to sneutrino relic from higgsino decay is negligibly small ( 2% − 3% of total Relic).
For the freeze-in via slepton and gaugino-dominated neutralinos, on the other hand, the relevant factor is the sneutrino mixing angle (θν). For a dominantly right-handed sneutrino dark matter, θν is given by, (3.5) Freeze-in constrains θν as a whole, and thus A ν , too, is subjected to constraints [24][25][26].
We We illustrate a few cases below, keeping all non-strongly interacting superparticle masses (excepting higgsino mass parameter µ) around the same scale, denoted by, m susy .
• For m susy 1 TeV we must have Γν DM 10 −21 GeV. As mentioned above we are looking for the least constrained scenario by choosing the lightest neutrino mass to be vanishingly small. Then the higgsino contribution to sneutrino relic is negligibly small and the main contribution comes from the decay of dominantly left-handed sfermions. The decay width of sfermions intoν DM is enhanced compared to those for gaugino and Higgsino decay by a phase-space factor m 2 susy m 2

EW
. For m susy 1 TeV this factor is 100. This in turn implies the mixing angle θν 3 × 10 −12 . Since θν is proportional to m −2 susy the numerator on the right-hand side of equation 3.5 has to be 3 × 10 −6 GeV 2 . There are two possible scenarios which give rise to such a small θν: -For a 'natural' scenario, each of the terms in the numerator of θν has to be 3 × 10 −6 GeV 2 . For Type I seesaw this implies that all the three Majorana neutrinos should be light, with M N O(100 MeV). This scenario is discussed in detail in section 4.1.
-A relatively fine-tuned situation occurs for M N 500 MeV or a few GeV. In this case, in order to obtain the correct relic density, one has to rely on the cancellation between µ v d and A ν v u . This situation is discussed in section 4.2.

Freeze-Out of NLSP
Till now we have considered only the contributions to sneutrino relic from heavier superparticles in thermal equilibrium. The NLSP is fairly long-lived, since its only possible decay intoν DM is suppressed by the small mixing angle θν. Thus, over and above the freeze-in process, a fair fraction of the NLSP population lives long enough to freeze-out of the thermal bath at an epoch to be decided by the Boltzmann equation [63,64]. The post freeze-out decay of NLSP also raisesν DM abundance to some extent. It should be noted here that since freeze-out is the only means of generating the DM abundance when the Yukawa coupling strength is very small, as happens for cases where neutrinos have just Dirac masses. The contribution to theν DM via NLSP freeze-out is given by [32,33] where Ω N LSP h 2 is the density of NLSP at freeze-out relative to critical density ρ c = 3 H 2 8 π G .
The contribution to relic density from the post freeze-out decay ofτ R NLSP, higgsino dominated χ 0 1 and bino dominated χ 0 1 NLSP are shown in figure 1. For the purpose of illustration we have kept SUSY breaking masses of left-slepton, wino around 1 TeV while the strongly interacting superparticle masses are kept in the range 1.2 TeV − 2.5 TeV in order to be consistent with recent LHC bounds while the pseudoscalar mass m A 0 is kept at 2.5 TeV along with tan β = 10.5. Such a choice of m A 0 prohibits higgsinos from reaching A-funnel region where its relic density is largely depleted. Thus throughout the region of M N LSP depicted in figure 1 higgsino relic are determined by coannihilation with lightest chargino (χ ± 1 ) and second lightest neutralino (χ 0 2 ). While for stau (τ R ) NLSP the relic density is determined by annihilation of staus into lepton pairs. The increase in the relic density with increase in mass of NLSP is attributed to decrease in s-channel annihilation cross-section. For a bino dominated χ 0 1 NLSP the relic density is determined by t-channel anihilation into lepton-antilepton pairs. With increase in bino mass parameter the corresponding anihilation cross-section increases quardratically with M N LSP and hence the relic density falls [65]. The figure 1 has been drawn with CP-evenν DM mass 97 GeV and CP-oddν DM mass 277 GeV. The NLSP abundance at freeze-out has been calculated using micrOmegas-5.0 [66].
Clearly for bino dominated NLSP the post freeze-out decay of NLSP overproducesν DM unless coanihilation comes into the picture. The situation is less binding for a Higgsinodominated NLSP as well as dominantly right-handed stau NLSP. Thus soft masses corresponding to left-sleptons, bino and wino are taken to be in the same scale in our analysis.

Constraints on light sterile neutrino
The active-sterile neutrino mixing angle, is given by It is severely constrained for a MeV-keV scale sterile neutrino as shown in figure 2 [67][68][69][70]. The constraint emerges for a sterile neutrino with MeV scale, since otherwise it decays after BBN and disturbes the light element abundance via its hadronic decay modes. This region is shown in yellow in figure 2. The constraints apparently disappears if the lifetime of the sterile neutrino is larger than the age of the Universe( 10 17 sec). However the mixing angle then suffers from a still stronger constraints from X-ray and gamma-ray observations attributed to the loop suppressed decay N R → ν L γ [41]. The violet region represents the parameter space thus ruled out. The orange region corresponds to the situation when the sterile neutrino abundance from the Dodelson-Widrow(DW) mechanism [37] is 1-100% of the currently measured DM relic density. The green region shows abundance of sterile neutrino exceeds the observed DM abundance. The red line in figure 2 denotes the mixing angle required to satisfy neutrino oscillation data via Type I seesaw mechanism. It is clear from the figure that for all three sterile neutrino masses 10 keV the mixing angle satisfying oscillation data is large enough to overproduce(Ω N h 2 >> Ω CDM h 2 ) the sterile neutrinos via the DW mechanism. Additional constraints on sterile neutrino masses other than what shown in figure 2 come from the Lyman-α forest observation [71][72][73] which eliminates M N 8 keV if the sterile neutrino is the only DM candidate. In our case sterile neutrino contribute only a fraction of the relic density, and hence the last mentioned   [38]. Barring the constraints from Big-Bang Nucleosynthesis (Yellow patch implies sterile neutrino lifetime is between BBN and the current age of the Universe) the MeV-keV scale sterile neutrino behaves as a decaying WDM candidate which can decay into an active neutrino along with a energetic photon and hence X-ray, Gamma ray observations restrict the associated parameter space severely (Violet) [41]. For low enough masses( 10 keV) the constraints on the present day energy density of the DM component of universe is proven to most stronger (Green). Orange patch signifies Dodelson-Widrow (DW) mechanism accounts for between 1% and all of dark matter (DM). red: mass-mixing angle combinations enforced by the seesaw mechanism. constraint is somewhat relaxed.

Big Bang Nucleosynthesis
The presence of long-lived particles is a potential threat to the light element abundance, as predicted by standard Big-Bang Nucleosynthesis (BBN). Long-lived particles may give rise to non-thermal nuclear processes (non-thermal BBN) attributed to their late decay into energetic SM particles. It disturbs the observed light element abundance. As mentioned earlier, we have considered the possibility of a right-handed stau (τ R ) and/or a neutralino(χ 0 1 ) as the NLSP. Aτ R NLSP can have late decay intoν DM and a W ± boson via the small neutrino Yukawa coupling. The worst victim of subsequent hadronic decays of the W ± is the deuterium abundance. A reasonable solution is to restrict the lifetime to O(100 sec) [14,50,[74][75][76][77], a constraint we have imposed through out on the spectrum. The decay χ 0 1 → νν DM leads to highly energetic light neutrinos. Their scattering with background neutrinos generate energetic e ± which subsequently produce energetic photons. These can change the light element abundances through photo-dissociation of light nuclei.
The three-body decay χ 0 1 →ν DM ν h ( * ) , though phase space suppressed, causes additional tension [78]. All such problems are avoidable by imposing an upper limit of O(100 sec) on the NLSP lifetime.

Other constraints
In addition to the aforementioned constraints, the following restrictions also have been adhered to in our numerical analysis: • The mass of lightest higgs emerging from the spectrum calculation modulo various uncertainties, is in the range 123 GeV < m h 0 < 128 GeV [79,80].

Results
We (d) All three Majorana masses are in the electroweak scale. In this scenario, say for m 1 10 −8 eV, the Yukawa coupling corresponding to the lightest neutrino will be ≈ 10 −10 and the freeze-in contribution from higgsino decay will overclose the Universe. However, we can always tune the lightest active neutrino mass to reduce the Yukawa coupling to a desired value and then the scenario is exactly similar to the preceding case. Hence we will not discuss this scenario in detail.
For cases (b) and (c) one right-handed Majorana mass is kept at the keV scale. In such cases, the neutrino warm DM and the non-thermal DM in the form ofν DM can coexist without being ruled out by freeze-in as well as freeze-out. However, it should be remembered that, in a hierarchical scenario, the lightest active neutrino mass being a free parameter the corresponding Majorana mass can in principle be made larger without changing the corresponding Yukawa coupling. All the interaction ofν DM depends on the order of its Yukawa coupling. In the first case, where all the Majorana masses are very small, the corresponding Yukawa couplings are also minuscule. Hence this scenario is very similar to the case of purely Dirac neutrinos. In the two remaining cases we have examined the effect of increase in the order of magnitude of Yukawa couplings on the freeze-in relic density ofν DM .

keV scale Majorana mass
In this case we have kept all the three Majorana masses (M N ) of neutrinos in the keV scale as mentioned already. We have confined ourselves to the following illustrative values of SUSY parameters: m susy = 1 TeV, tan β = 10.5.
The largest Yukawa coupling for keV scale Majorana neutrinos is O(10 −10 ) and hence the term v u √ 2 M N Y ν in the numerator of sneutrino mixing angle, θν, is 10 −13 GeV 2 .
This is much smaller than that required for m susy 1 TeV as discussed in section 3.1. Hence the relic density is determined by the term v u √ 2 (A ν − µ cot β) Y ν . Thus as the difference between A ν and µ cot β increases, the freeze-in relic density also goes up. The variation of freeze-in relic density with µ for three different values of A ν with M N = 10 keV is shown in figure 3 (left). As is clearly seen there for A ν = 100 GeV the relic density is minimum around µ 1.05 TeV. As one departs from this value of µ, the relic density increases as a higher θν enhances the freeze-in process. The smooth increase in relic density is accounted for the low value of Y ν multiplied to ( does not play a significant role in this case, when A ν and µ are tuned to cancel upto five decimal places. A similar feature is also seen for A ν = 50 GeV. The relic density for A ν = 20 GeV is minimum for µ = 210 GeV. Similarly, for fixed µ and A ν , the relic density increases with the Majorana mass due to the term vu

Majorana masses in the scale MeV-GeV
It is evident from figure 2 that, the simultaneous fulfillment of oscillation data, neutrino relic density constraints and BBN requirements imply that the two heavier sterile neutrinos must have masses M H N 500 MeV. For such a choice of parameters the largest Yukawa coupling is O(10 −8 ) and hence the term vu √ 2 M N Y ν 10 −6 GeV 2 , which is what necessary to obtain correct relic density following our discussion in section 3.1. However, one would also require the term v u √ 2 (A ν − µ cot β) Y ν to be on the same order of magnitude, namely, O(10 −6 ) GeV 2 . For A ν and µ in the electroweak scale, this is only possible if these two terms cancel between themselves. Thus one needs A ν µ cot β , (4.1) in order to reproduce the correct relic density. Figure 4 (left) shows the variation of relic density with A ν for µ = 800 GeV and tan β = 10.5. One can see that for µ = 800 GeV and tan β = 10.5 relic density is minimized at A ν = 76.2 GeV which is in agreement with equation 4.1. The situation is quite similar for µ = 1200 GeV as shown in figure 4 (right). Figure 5 depicts the variation of relic density with µ for A ν = 100 GeV, which minimizes around µ = 1.05 TeV.
One must also notice that in situations when the two terms µ cot β and A ν cancels exactly, the mixing angle θν is dominated by the term v u √ 2 M N Y ν . This explains the increase in relic density with the heavier Majorana masses(M H N ). Thus, for a fixed values µ and tan β, the required A ν to reproduce correct relic density is also constrained. The . Hence the term v u √ 2 M N Y ν is 10 −3 GeV 2 , which is about 3 × 10 3 times larger than the value required for allowing m susy 1 TeV. Hence with electroweak scale Majorana neutrinos TeV scale SUSY breaking is disfavored for a right-handed sneutrino LSP. To reconcile electroweak scale Majorana masses with right sneutrino LSP, one has to increase the SUSY breaking scale to a much higher value (m susy 100 TeV) since the sneutrino mixing angle decreases quadratically as m susy increases. In figure 6 the dependence of relic density on the variation of SUSY breaking scale is shown for µ and A ν both around EW scale with tan β = 3.5. It should be noted that, in such a case, the freeze-in constraint on θν requires A ν and Mν DM to be still within a TeV. On the other hand, µ should also be within a TeV, not only for satisfying electroweak symmetry breaking conditions, but also to control freeze-in viaν L →ν R h. However, we then face an inexplicable hierarchy of SUSY-breaking parameters, unless one engineer some highly contrived fine-tuning.
The monotonic fall in relic density with increasing m susy can be attributed to the dependence of θν on 1/m 2 susy . Another interesting point to note is that the relic density for µ = 800 GeV is larger for A ν = 500 GeV compared to, say, A ν = 100 GeV while the situation is opposite for µ = 1500 GeV. In case of µ = 800 GeV the term (A ν − µ cot β) is larger for A ν = 500 GeV compared to A ν = 100 GeV, thereby enhancing θν. The situation is exactly opposite for µ = 1500 GeV.

Quasi-degenerate neutrino masses
It has been already mentioned that, for hierarchical neutrino masses, the free parameter in the form of the lightest active neutrino mass may be tuned to control the sneutrino abundance. For the quasi-degenerate scenario, the lightest neutrino mass is bound to be 10 −2 eV. In order to prevent higgsino decays from overproducing sneutrino dark matter via Yukawa interaction, one has to lower the Majorana mass corresponding to the lightest neutrino eigenstate. One thus finds oneself pushed down to eV-scale value for this Majorana mass. Such a situation has already been mentioned in [34]; however, recent limits from BBN and recombination [45,50] strongly restricts this scenario. Although some wisely engineered ways of evading such constraints exist [82][83][84][85][86][87][88], we do not consider this possibility in greater detail.

Summary and conclusions
We have worked with an MSSM scenario augmented with three families of right-chiral neutrino superfields, and ∆ L = 2 terms in the superpotential as well as the scalar potential where masses for the light neutrinos have been generated following Type-I seesaw mechanism. The right-handed sneutrinos are non-thermal dark matter candidate as a result of small Yukawa couplings. We have restricted to hierarchical neutrino masses, latest constraints from Planck data on DM as well as data on the neutrino sector. We have gone beyond earlier studies where constraints on the neutrino Majorana masses were obtained, mostly within a degenerate neutrino scenario. While the eV-scale scenario proposed there is presently disfavoured, we have shown that the current picture admits of considerable varieties as well as constraints in view of current observations. These include, on the theoretical side, constraints from both the NLSP freeze-out and freeze-in of right sneutrinos as well as sterile neutrinos. Our conclusions are: • While the lightest neutrino mass and consequently the corresponding entry in the (diagonal) right-handed neutrino mass matrix is a free parameter, the remaining ∆ L = 2 mass terms are constrained rather tightly.
• ∆ L = 2 masses on the electroweak scale and above are possible only if there is a hierarchy of 2-3 orders of magnitude between DM mass Mν DM , µ and A ν on the one hand, and the remaining SUSY-breaking masses on the other.
• For ∆ L = 2 masses in the range 500 MeV-a few GeV, all constraints can be satisfied only if A ν and the µ-parameter are fine-tuned to 4-5 decimal places.
• Majorana masses in the few keV-500 MeV range are disfavoured by standard BBN, as the sterile neutrinos have lifetimes too long to maintain the observed light element abundances.
• If all three ∆ L = 2 entries in M N are in the keV range, then all three sterile neutrinos effectively constitute warm dark matter. Such a situation, however, is disfavoured by the Dodelson-Widrow mechanism where the freeze-in rate of the warm DM goes up unacceptably.
We thus conclude that the right-sneutrino DM scenario, if at all the picture of nature, is subject to rather severe constraints for practically all orders of magnitude of the ∆ L = 2 masses. The exception to this requires an inexplicable hierarchy among SUSY-breaking mass parameters which perhaps can be justified by going beyond MSSM.

A Decay Widths
The decay widths of the channels that contribute to freeze-in ofν DM are as follows [89]:

B Amplitudes
In this section we give the expressions for the amplitudes for all the decays considered in section 3.1. In the expressions that follows we have denoted the U P M N S as U 0 and sneutrino mixing matrix as Zν. U V will denote the full 6 × 6 matrix that diagonalizes the 6 × 6 neutrino mass matrix given by, The index i , j denotes the superparticle generation appearing in the vertex while I denotes the same for R-parity even particles. Forν DM the index i = 6 need to be used.

B.2 Interactions with sfermions
Sneutrino LSP can decay from sleptons and heavier sneutrinos via SM gauge bosons or SM higgs which we have discussed below: 2. For a heavier CP-even(or CP-odd) sneutrino(ν j H ) decaying into a lighter CP-odd(or CP-even) sneutrino (ν i ), For higgs-mediated sneutrino decays also j runs from 1 to 5 as in the case of Zmediated processes.
In diagonalizing sneutrino mass matrix and calculating all the Feynman Rules we have used SARAH [90].