Sterile neutrinos

Neutrino masses imply a likely existence of gauge-singlet fermions, which may be very heavy but may also be light. In the latter case, sterile neutrinos appear in the low-energy theory. They can make up all or part of dark matter and can explain the observed velocities of pulsars because they are emitted anisotropically from a cooling neutron star born in a supernova explosion. The most promising detection opportunity is by using x-ray telescopes and searching for monochromatic x-ray photons emitted from decays of these particles in dark-matter-dominated halos.


Introduction
The Standard Model was originally formulated with massless neutrinos, which were included as the only left-handed fermions that had no right-handed counterparts. The supposed masslessness of neutrinos was, thus, explained by the absence of corresponding singlets and by lepton number conservation. The discovery of neutrino masses [1] can be easily implemented by a small modification of the Standard Model if the right-handed SU (2) singlets are added and supplied with the mass terms. The right-handed gauge-singlet fields can, however, have Majorana masses as well. These Majorana mass terms can be much heavier than the electroweak scale, in which case the corresponding additional degrees of freedom disappear from the low-energy effective Lagrangian. However, if the Majorana masses are below the electroweak scale, the new light degrees of freedom can be observed at low energy, and can account for dark matter [2] and pulsar kicks [3]- [5]. These light fermions, coupled to weak currents only via mixing with active neutrinos, are called sterile neutrinos 1 . The search for sterile neutrinos is motivated by the fact that active neutrinos have mass, as well as by astrophysical data that can be explained by sterile neutrinos.

Neutrino masses and sterile neutrinos
The easiest way to introduce neutrino masses is by using n electroweak-singlet fermions N a (a = 1, . . . , n) in the seesaw Lagrangian [7]- [11]: Here L SM is the Standard Model Lagrangian (with only left-handed neutrinos and without neutrino masses). Active neutrinos ν α are included as components of the electroweak SU(2) doublets L α (α = e, µ, τ ). We assume that SU(3)-triplet Higgs bosons [12] are not involved, and all neutrino masses arise from Lagrangian (1), while the higher-dimensional operators are not important. This Lagrangian, obviously, has additional degrees of freedom as compared with the Standard Model. The corresponding particles can be very heavy or, if some of them are light, can play the role of dark matter. The lowest number of singlets consistent with neutrino masses is n = 2 [13]. With n = 3 singlets, one of which has a mass of several keV and the other two have closely degenerate masses around several GeV, in the model called νMSM, one can fit the neutrino masses and explain dark matter and baryon asymmetry of the universe (see [14] and references therein).
The neutrino mass eigenstates ν (m) i (i = 1, . . . , n + 3) are linear combinations of the weak eigenstates {ν α , N a }. They are obtained by diagonalizing the (n + 3) × (n + 3) mass matrix: If one assumes y aα H ∼ y H M a ∼ M, the eigenvalues of this matrix split into two groups: lighter states with masses 3 and heavier eigenstates with masses of the order of M: The former are called active neutrinos and the latter are called sterile neutrinos.
It is often assumed that M 1 TeV, but the naturalness arguments can be given in favor of either the large or the small value of M [15,16]. It is also possible that some Majorana masses are much heavier than the others. If at least one of them is at the electroweak scale, the light sterile neutrinos appear in the low-energy spectrum.
The Majorana mass can also be generated by the Higgs mechanism, just like the masses of the other light fermions. This requires the presence of a singlet S in the Higgs sector. Then interaction of the form where S can be either real or complex, can generate the Majorana mass after the boson S acquires a vacuum expectation value (VEV). This coupling opens a new channel for the production of relic sterile neutrinos in the early universe, S → N N , which is independent of the mixing angles and can operate at a high temperature, which results in a colder population of dark matter [17]- [21].

Dark matter in the form of sterile neutrinos
Sterile neutrinos with masses of the order of several keV, having a small flavor mixing with active neutrinos, can account for all or part of cosmological dark matter. Dodelson and Widrow [2] realized that such sterile neutrinos would be out of thermal equilibrium at all temperatures in the early universe and that they would be produced via active-sterile neutrino oscillations [2], [22]- [32]. The production of sterile neutrinos could be enhanced by the resonance due to some nonzero lepton asymmetry [22,28,32]. Finally, sterile neutrinos could be produced by some mechanisms that do not involve oscillations, such as inflaton [33] or Higgs decays [19,20,33]. Dark matter in the form of sterile neutrinos has a non-negligible free-streaming length, which depends on the production mechanism as well as on mass. This dark matter fits well the structure on large scales, and yields predictions on small scales [17,18,21] that will be tested by observations in the near future by various astrophysical techniques [34]- [41].
If the Majorana mass is generated by the VEV of the singlet Higgs, and if one requires that the VEV and the mass of this singlet Higgs boson be of the same order of magnitude (a natural assumption), then there is an intriguing relation between the mass and VEV of the S boson, the Yukawa coupling required to produce the right amount of dark matter and the mass of the sterile neutrino [19,20]. If two of these three parameters are specified, the third is a nontrivial prediction of the model (which agrees with the data).

Pulsar kicks
There is an intriguing independent hint in favor of the existence of the sterile neutrino with properties that are consistent with it being dark matter. The emission of these particles from a cooling neutron star would be anisotropic and could manifest itself in a large recoil velocity of the neutron star. Neutron stars, observed as pulsars, are known to have large velocities, the 4 origin of which remains unclear (see [42] and references therein). Numerical simulations of the supernova explosion are not in agreement [43,44], although some have claimed to produce a strong enough kick [45]. Some aspects of the pulsar data are difficult to understand in connection with these results. For example, if the kick is generated by the spiral mode of the spherical accretion shock instability [46], one would expect the kick in the direction orthogonal to the axis of rotation, while the data indicate the − v correlation [47]- [49].
Although both pulsar kicks and dark matter can be explained as due to some other reasons, unrelated to sterile neutrinos, it is intriguing that keV sterile neutrinos explain both. In other words, if a particle with a mass of several keV and a small mixing exists, the existence of dark matter and the large velocities of pulsars would be the consequence.
The reason why sterile neutrinos can explain pulsar kicks is their anisotropic emission from the cooling neutron star. To account for the observed velocities of pulsars, an asymmetry in neutrino emission as small as 1% is sufficient.
Even the ordinary, active neutrinos are produced in processes that are anisotropic because of the strong magnetic field in a neutron star. The neutrinos do not interact with the magnetic field directly (their magnetic moments are small), but they are produced in Urca reactions such as n + e + p +ν e , which involve electrons and protons, whose spins are aligned along the magnetic field. Indeed, the probability to produce a neutrino along the magnetic field is not the same as that in the opposite direction: Depending on the fraction of electrons in the lowest Landau level, this asymmetry can be as large as 30%, which is, seemingly, more than that needed to explain pulsar kicks. However, this asymmetry is completely washed out by the scattering of neutrinos on their way out of the star [50,51]. This is intuitively clear because, as a result of scattering, the neutrino momentum is transferred to and shared by the neutrons. The neutrinos undergo multiple scattering and remain almost in equilibrium as they diffuse out of the protoneutron star. In approximate thermal equilibrium, no asymmetry in the production or scattering amplitudes can result in macroscopic momentum anisotropy 2 However, the sterile neutrinos, produced in the same interactions with the cross sections suppressed by the square of the mixing angle, undergo no scattering on their way out of the neutron star. They are produced with anisotropy, which can be as high as 10-30%, and they escape with the same anisotropy. Therefore, if sterile neutrinos carry a fraction (0.03-0.1) of the total supernova energy, they generate the 1% anisotropy required to explain the pulsar kicks [4,5]. In addition, a Mikheev-Smirnov-Wolfenstein [52,53] resonance may occur at some density and can cause a kick due to a somewhat different mechanism [3,54,55]. This mechanism can be tested using observations of radio pulsars and supernova remnants. One prediction is that the kick velocities should be aligned with the axis of rotation of the neutron star, in agreement with recent observations [47]- [49]. Another prediction is that asymmetric jets should have an asymmetry with a stronger jet pointing in the direction of the neutron star motion [44]. The allowed parameter space has a significant overlap with that allowed for dark matter, as shown in figure 1. The excluded region (solid filled) is based on the assumption that neutrino oscillations are the only source of sterile neutrinos. This is the only production mechanism that, given the standard cosmological history, is independent of model assumptions. However, the amount of dark matter produced in this way may be too small. If some additional mechanisms make up the difference, and all the dark matter is composed of sterile neutrinos, the exclusion region is to the right of the solid line. The regions favored by pulsar kicks are also shown.

The search using Suzaku and other x-ray telescopes
Since the mixing angles of dark-matter sterile neutrinos must be very small (figure 1), they are unlikely to be detected in laboratory experiments. However, large concentrations of darkmatter particles in halos offer an opportunity to detect them using the one-loop decay process: ν s → γ ν a . This is a two-body decay, and the predicted signal must, therefore, be a spectral line with energy equal to half the particle mass. The current exclusion plot based on a number of observations [24], [56]- [74], including the search using the Suzaku x-ray telescope [72,75], is shown in figure 1.

Effects of x-rays during the dark ages
If dark matter is made up of sterile neutrinos, their decay during the 'dark ages' can affect the formation of the first stars [76]- [80]. Initial star formation depends on the fraction of molecular hydrogen, which can increase due to the effects of ionizing radiation produced by sterile neutrino decays. Since there are fewer small halos in the case of warm dark matter, the formation of first stars starts somewhat later, but proceeds faster because of the ionizing radiation from dark-matter decay [76,77]. Hence, the overall effect is to make star formation more prompt, occurring in the narrower range of redshifts. Future observations may be able to test this prediction.

Conclusion
Sterile neutrinos present a plausible dark-matter candidate. They are motivated by the discovery of neutrino masses: the seesaw Lagrangian that includes singlet fermions predicts the existence of sterile neutrinos if some of the Majorana masses are small. There are independent hints in favor of such particles: they would be emitted anisotropically from a cooling neutron star born in a supernova explosion. The asymmetry would be strong enough to give the pulsar a recoil velocity as high as 10 3 km s −1 , which is consistent with observation. Future observations of x-ray telescopes can help confirm or rule out the sterile neutrino as a dark-matter candidate.