The νMSM, Dark Matter and Neutrino Masses

Abstract We investigate an extension of the Minimal Standard Model by right-handed neutrinos (the νMSM) to incorporate neutrino masses consistent with oscillation experiments. Within this theory, the only candidates for dark matter particles are sterile right-handed neutrinos with masses of a few keV. Requiring that these neutrinos explain entirely the (warm) dark matter, we find that their number is at least three. We show that, in the minimal choice of three sterile neutrinos, the mass of the lightest active neutrino is smaller than O(10−5) eV, which excludes the degenerate mass spectra of three active neutrinos and fixes the absolute mass scale of the other two active neutrinos.

Introduction.-In the past decade, neutrino experiments have provided convincing evidence for neutrino masses and mixings. The anomaly in atmospheric neutrinos is now understood by ν µ → ν τ oscillation [1], while the solar neutrino puzzle is solved by the oscillation ν e → ν µ,τ [2,3] incorporating the MSW LMA solution [4]. Current data are consistent with flavor oscillations between three active neutrinos [5], and show that the mass squared differences are ∆m 2 atm = [2.2 +0.6 −0.4 ] · 10 −3 eV 2 and ∆m 2 sol = [8.2 +0.3 −0.3 ] · 10 −5 eV 2 [8]. These phenomena demand physics beyond the minimal standard model (MSM), and various possibilities to incorporate neutrino masses in the theory have been proposed [9]. The simplest one is adding N right-handed SU(2)×U(1) singlet neutrinos N I (I = 1, . . . , N ) with most general gauge-invariant and renormalizable interactions described by the Lagrangian: where Φ and L α (α = e, µ, τ ) are the Higgs and lepton doublets, respectively, and both Dirac The νMSM with N singlet neutrinos contains quite a number of free parameters, i.e.
Dirac (M D I,α ) and Majorana (M I ) masses. For example, for N = 2 the number of extra real parameters is 11 (2 Majorana masses, 2 Dirac masses, 4 mixing angels and 3 CP-violating phases), whereas for N = 3 this number is 18 (3 Majorana masses, 3 Dirac masses, 6 mixing angels and 6 CP-violating phases). These parameters can be constrained by the observation of neutrino oscillations. The immediate consequence of the existence of two distinct scales ∆m 2 atm and ∆m 2 sol is that the number of right-handed neutrinos must be N ≥ 2. However, we know little about the absolute values of masses for active neutrinos as well as righthanded neutrinos. This is simply because the oscillation experiments tell us only about the mass squared differences of active neutrinos.
On the other hand, cosmology can play an important role to restrict the parameter space of the νMSM. Recently, various cosmological observations have revealed that the universe is almost spatially flat and mainly composed of dark energy (Ω Λ = 0.73 ± 0.04), dark matter (Ω dm = 0.22 ± 0.04) and baryons (Ω b = 0.044 ± 0.004) [11]. The νMSM can potentially explain dark matter Ω dm and baryon Ω b abundances, and can be consistent with the dark energy requirement via the introduction of a small cosmological constant.
To be more precise, the baryon asymmetry of the universe (Ω b ) can be produced via the leptogenesis mechanism [12] or via neutrino oscillations [13] with the use of anomalous electroweak fermion number non-conservation at high temperatures [14]. Furthermore, the νMSM can offer a candidate for dark matter. The present energy density of active neutrinos is severely constrained from the observations of the large scale structure. The recent analysis [15] shows that the sum of active neutrino masses should be smaller than 0.42 eV and Ω ν h 2 ≤ 4.5 · 10 −3 , which is far below the observed Ω dm . The unique dark-matter candidate in the νMSM is then a right-handed neutrino which is stable within the age of the universe.
Indeed, it has been shown in [16] - [20] that sterile right-handed neutrinos with masses of O(1) keV are good candidates for warm dark matter. Note that, in our analysis, we take the very conservative assumption of the validity of the standard Big Bang at temperatures below 1 GeV and disregard the possibilities of extremely low reheating temperatures of inflation as T R < ∼ 1 GeV [21].
In this letter, we explore the hypothesis that the νMSM is a correct low-energy theory which incorporates dark matter. We demonstrate that the theory with N = 2 fails to do so.
We show that for the choice N = 3 the mass of the lightest active neutrino m 1 is constrained from above by the value O(10 −5 ) eV, and therefore, that the masses of other neutrinos are fixed to be m 2 = ∆m 2 sol and m 3 = ∆m 2 atm + ∆m 2 sol in the normal or m 2 = ∆m 2 atm and m 3 = ∆m 2 atm + ∆m 2 sol in the inverted hierarchy of neutrino masses, respectively. This rejects the possibility that all active neutrinos are degenerate in mass. In other words, for a most natural choice of N = 3, the cosmological observation of dark matter allows one to make a (potentially) testable prediction on the active neutrino masses and on the existence of a sterile neutrino with a mass in the keV range. We stress that these results are valid in spite of a large number of free parameters of the νMSM. Finally, for N ≥ 4, no model-independent extra constraints on the masses of active neutrino can be derived.
Neutrino Masses and Mixing.-Let us first discuss neutrino masses and mixing in the νMSM. We will restrict ourselves to the region in which the Majorana neutrino masses are larger than the Dirac masses, so that the seesaw mechanism [22] can be applied. Note that this does not reduce generality since the latter situation automatically appears when we require the sterile neutrinos to play a role of dark matter, as we shall see. Then, right-handed neutrinos N I become approximately the mass eigenstates with M 1 ≤ M 2 ≤ . . . ≤ M N , while other eigenstates can be found by diagonalizing the mass matrix: which we call the seesaw matrix. The mass eigenstates and the mixing in the charged current is expressed by ≪ 1 under our assumption. This is the reason why right-handed neutrinos N I are often called "sterile" while ν i "active".
As explained before, ∆m 2 atm and ∆m 2 sol require the number of sterile neutrinos N ≥ 2. For the minimal choice N = 2, one of the active neutrinos is exactly massless (m 1 = 0). For A sterile neutrino, say N 1 , decays mainly into three active neutrinos in the interesting mass range M 1 ≪ m e (see Eq. (7) below) and its lifetime is estimated as [19] where we have taken |Θ α1 | = Θ for α = e, µ, τ . We can see that it is stable within the age of the universe ∼ 10 17 sec in some region of the parameter space (M 1 ,Θ).
When the active-sterile neutrino mixing |Θ αI | is sufficiently small, the sterile neutrino N I has never been in thermal equilibrium and is produced in non-equilibrium reactions. The production processes include various particle decays and conversions of active into sterile neutrinos (see Ref. [23]). The dominant production mechanism is due to the active-sterile neutrino oscillations [17,19,20], and the energy fraction of the present universe from the sterile neutrino(s) is [20] Ω N h 2 ∼ 0.1 where the summation of I is taken over the sterile neutrino N I being dark matter. The most effective production occurs when the temperature is T * ≃ 130 MeV(M I /1 keV) 1/3 [17,24].
Here we assumed for simplicity the flavor universality among leptons in the hot plasma, which is actually broken since T * ≤ m τ . However, its effect does not alter our final results.
Further, we have taken the lepton asymmetry at the production time to be small (∼ 10 −10 ), which is a most conservative assumption. In this case there is no resonant production of sterile neutrinos coming from large lepton asymmetries [18,20]. We therefore find from the definition of Θ that the correct dark-matter density is obtained if where m 0 = O(0.1) eV. Notice that this constraint on dark-matter sterile neutrinos is independent of their masses, at least for M I in the range discussed below.
The sterile neutrino, being warm dark matter, further receives constraints from various cosmological observations and the possible mass range is very restricted as where the lower bound comes from the cosmic microwave background and the matter power spectrum inferred from Lyman-α forest data [25], while the upper bound is given by the radiative decays of sterile neutrinos in dark matter halos limited by X-ray observations [26].
These constraints are somewhat stronger than the one coming from Eq. (4).

Consequence of Sterile Neutrino Dark
Matter.-We have found that the hypothesis of sterile neutrinos being warm dark matter is realized in the νMSM when the two constraints (6) and (7) are satisfied. We shall now see that they put important bounds on the number of sterile neutrinos and on the masses of the active ones. To find them, let us first rewrite the diagonalized seesaw mass matrix (3) in the form where S I denotes a contribution from each sterile neutrino and is given by (S I ) ij = X Ii X Ij with X Ii = (M D U) Ii / √ M I . Note that each matrix satisfies the relation det S I = det(S I + S J ) = 0 from its construction. The condition (6) is then written as and the mass range in Eq. (7) gives First of all, let us show that the minimal possibility N = 2 cannot satisfy the dark-matter constraints and the oscillation data simultaneously. In this case, the lightest active neutrino becomes massless (m 1 = 0). By taking the trace of both sides in Eq. (8), we find that This equation must hold for both real and imaginary parts. When both sterile neutrinos N 1 and N 2 are assumed to be dark matter, the condition (9) together with M 1 and M 2 in Eq. (7) leads to This inequality cannot be satisfied since m dm ν = O(10 −5 ) eV and m 3 = ∆m 2 atm + ∆m 2 sol ≃ 5 · 10 −2 eV from neutrino oscillations.
Further, when only one of two sterile neutrinos, say N 1 , is assumed to be dark matter, its Dirac Yukawa couplings are restricted as shown in Eq. (9). Although the couplings of The vanishing determinant of the second matrix on the right-hand side leads to By taking into account the dark matter constraint |X 12 | 2 + |X 13 | 2 = m dm ν , we obtain the upper bound on m 2 : This inequality is inconsistent with m dm ν in Eq. (10) and m 2 = ∆m 2 sol ≃ 9 · 10 −3 eV or ∆m 2 atm for the normal or inverted hierarchy cases, respectively. The same discussion can be applied to the case when only the heavier sterile neutrino N 2 is dark matter. Therefore, we have shown that in the N = 2 νMSM the requirements on dark matter conflict with the oscillation data.
We then turn to discuss the case N = 3. First, when all three sterile neutrinos play a role of dark matter simultaneously, the real part of the trace of Eq. (8) gives where the final inequality comes from the dark matter constraint (9) as in the previous case.
Although we do not know the overall scale of m i from the oscillation data, the heaviest one m 3 should be larger than ∆m 2 atm in any case. Then, this inequality cannot be satisfied by m dm ν in Eq. (10) and this situation is excluded.
Next, we consider the case when two of the three sterile neutrinos, say N 1 and N 2 , are dark matter. In this case, from the real part of the trace of Eq. (8), we find that and thus ReX 3i 2 > m 3 since m dm ν ≪ ∆m 2 sol ≤ m 2 . On the other hand, it is found from However, this equation cannot be satisfied, since the real part of the right-hand side is bounded from below as If m 1 = 0, det(M ν diag − S 3 ) = 0 gives us X 31 = 0. This results in that M ν diag and S 3 as well as (S 1 + S 2 ) are reduced to 2 × 2 matrices, which verify det S 3 = det(M ν diag − S 1 − S 2 ) = 0, i.e.
This equation cannot be satisfied by X Ii restricted by the dark matter constraint (9). Thus, this case is also excluded in either m 1 = 0 or m 1 = 0 situations.
Finally, let us consider the remaining possibility, i.e. assume that only one sterile neutrino (e.g. N 1 ) becomes a dark matter particle. In this case, we also note that det(S 2 + S 3 ) = det(M ν diag − S 1 ) = 0, which induces Now, the dark matter constraint (9) takes the form: It is then found that the lightest active neutrino should verify matter would be the discovery of a keV sterile neutrino by the X-ray observatories [26] and the finding of the active neutrino masses in the predicted range.
Finally, we should mention that the sterile neutrinos irrelevant to dark matter can be responsible for the baryon asymmetry of the universe through leptogenesis [12] or neutrino