Interactions of MeV and GeV sterile neutrinos with matter

The discovery of neutrino masses suggests the likely existence of gauge singlet fermions (right-handed neutrinos) that participate in the neutrino mass generation [1]. A sterile neutrino is a hypothetical neutrino that does not interact via any of the fundamental interactions of the Standard Model except gravity. It is a right-handed neutrino or a left-handed anti-neutrino. Such a particle belongs to a singlet representation with respect to the strong interaction and the weak interaction and has zero weak hypercharge, zero weak isospin and zero electric charge. Sterile neutrinos would still interact via gravity, so if they are heavy enough, they could explain cold dark matter or warm dark matter. The X-ray observations make use of the radiative decay of a sterile neutrino [2,3], can yield a non-negligible flux from concentrations of dark matter in astrophysical systems, such as, e.g., galaxies, clusters, and dwarf spheroidal galaxies [1,4]. The photons emitted from decays sterile neutrinos can affect the formation of the first stars. Their production in a supernova can also explain the pulsar kicks and they have many other implications in astrophysics and cosmology. It is of interest, therefore to study the interactions of sterile neutrinos in matter with the purpose of possibly using them to inform the direction of current and future experimental searches. 2 Interaction with atoms and ionization of atom.

In this section we study the scattering of a MeV sterile neutrino by an electron: ν s e -→ ν e e -. The relevant Feynman diagram depicted in Fig. 1. The effective Hamiltonian for this scattering process is the same as ordinary neutrino-electron scattering and is given by [5]:

Where:

2 + 2 sin 2 θ W , and sin 2 θ W = 0.23 is the weak-mixing angle. To calculate the unpolarized cross section which means that no information about the electron spins is recorded in the experiment and to allow for scattering in all possible spin configurations, we make the following replacement:

Where s A , s B are the spins of the incoming particles. That is, we average over the spins of the incoming particles and we sum over the spins of the particles in the final state. Then we have:

Where s = (p νs + p e ) 2 , t = (p νe -p νs ) 2 , and u = (p e -p νe ) 2 are the Mandelstam variables. To calculate |M | 2 , we should first calculate the Mandelstam variables s, t, u. since m νs ∼ = 5keV [5], the sterile neutrinos are relativistic which means that one can set m νs ∼ = 0. Therefore the variables s, t, u take the following forms : s ≈ m 2 e + 2m e E νs , t ≈ -2E νe E νs (1 -cosθ ν ), u ≈ m 2 e -2m e E νe , Where θ ν is the scattering angle of the final neutrino ν e . we may write the differential cross section, in the symbolic form :

Where dQ is the invariant phase space factor and F is the initial flux. By calculating |M |

, F and dQ, we obtain the following expression for the scattering cross section :

Where υ is the relative velocity before scattering. Let us now study the interaction of a GeV sterile neutrino with an atom. Since the energy of the sterile neutrino is much more than the ionization energy and since the rest mass energy of the electron is 0.5 MeV,the final electron is relativistic, so we can set m e ∼ = 0. The sterile neutrino is also relativistic in the GeV energy range. The scattering cross section is as follows :

We now compare the result of ionization of atoms by keV and MeV-GeV sterile neutrinos. The momentum transfer to the electon for the keV sterile neutrino [5] is |p e | = m νs and the electron kinetic energy in the final state is > 25 eV and in the GeV case T e » 25 eV. In MeV case the energy is more than needed to ionize the atom and for the GeV sterile neutrinos it is much more than the ionization energy. Therefore if one can measure the electron spectrum resulting from these interactions it would peak at the energies more than or much more than 25 eV. As mentioned in [5] the well-defined prediction for the energy spectrum can be useful to reject potential backgrounds. In MeV and GeV energy ranges the scattering of a sterile neutrino off the electrons in an atom is not coherent. Because the momentum transfer is more than m νs ≈ 5 keV, so it corresponds to the compton wavelengh smaller than the size of atom, 10 -8 cm, the sterile neutrino does not scatter coherently of all the electrons in the atom, therefore we take σ νsA ≈ σ νse .

Let us compare the event rate of ν s A scattering in keV and MeV-GeV energy ranges. It is given by R νsA = σ νsA υn νs N T , where n νs is the number density of sterile neutrino dark matter, and N T is the number of target atom A in the detector. As [5], we assume that the local mass density of dark matter is 0.4 GeV cm -3 , and it is only made of sterile neutrinos, their number density is n νs = 8 × 10 4 cm -3 (m νs /5keV ) -1 . The number of target atoms is N T = (6 × 10 29 /A)(M det /ton), where M det is the mass of the detector. It is shown that [1,2] the life

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