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Neutrino-bound di-neutrons as an exotic metastable atom

It is shown the possibility of existence of metastable atoms of dineutroneum as a bound state of two neutrons and one neutrino. Such atoms can appear in a reaction of deuterons with free or quasi-free electrons. The estimation of mass, size and lifetime of dineutroneum atom is fulfilled.


Introduction
Laws of physics do not impose basic theoretical bans on the existence of the metastable bound state of the two neutrons and neutrino [1], because a neutrino is a massive particle [2].  [3,4].
Due to interaction with quarks in a nucleon, a neutrino can "linger" inside it.
This delay is caused, because the effective N ν -potential corresponding to Wboson exchange (Fig. 1), is a short-range and very deep one. Its depth is still rather small to keep antineutrino, proton and electron in the bound state (i.e. like a neutron) for a long time, but just enough to consider a proton like the stable bound state of three particles, positron, neutron and neutrino. It is well known, that three-body effects allow an existence of 3 particles' bound states, which pair potentials are insufficiently deep to form 2 particles' bound states.
A long lifetime of the neutrino inside a nucleus can be treated on the basis of exotic Miheev -Smirnov -Volfenstein effect at low energies [5]. Let us explain this in more detail. If the energy of incoming electron is resonant (i.e. renormalized masses of all three types of neutrinos ( , , e µ τ ν ν ν ) inside a nucleon are approximately equal after the electron capture), the exotic nucleus is generated at the first stage of electroweak process (two left vertexes in the diagram 1), which cannot decay until an oscillation have been finished. The exotic nucleus D ν is metastable, because the energy conservation law forbids its decay with µ -or τ -lepton emission. The From this standpoint, we shall consider hypothetical metastable exotic atom (exotic nucleus) dineutroneum, which is the bound state of two neutrons and one neutrino, as was mentioned above. The aim of this work is to estimate the mass, size and lifetime of the dineutroneum atom which is formed due to the interaction of deuterons with free, or quasi-free electrons.

Main formalism
The known Hamiltonian of weak interaction is with G constant of universal weak interaction Fermi, ( ) J r λ weak current, and ˆ( , ) G r r ′ propagator. Let us introduce definition in accord to [6] ( ) and similarly for others 4-vector operators. In the standard model, the weak interaction is caused by exchange of the W -boson with mass ~90GeV . Therefore, if we consider the low energy weak processes, an approximation W m → ∞ can be used. Accordingly, the interaction is quite local, and components of the weak current in Hamiltonian (1) should be taken at the same point of space The Lorenz invariant weak current is well known. For example, β -decay of a neutron is described by the Hamiltonian [6] To describe the weak processes in nuclear physics, one needs in a non-relativistic The model of the Hamiltonian was derived in the early papers of Fermi, Gamov and Teller, and looks like [6] { } the lepton current, E the energy positive for particles and negative for antiparti- In the works devoted to the nuclear β -processes, the WFs of free leptons in (6) are usually chosen as plane waves with the momentum l p 1 . Thus, the lepton's current (6) looks like: where k e ν = − the lepton transferred momentum, ν the wave vector of neutrino, e the wave vector of electron, 3 L the normalization volume, ( ) The spinor 1/ 2 m ν = ± the spin projection of neutrino ( + corresponds to spin "up" and − spin "down").
The lifetime of dineutroneum can be estimated within the approximation of allowed transitions. Therefore, we shall neglect the small contribution of the terms /( ), / k Mc p Mc и kR due to the forbidden transitions, and obtain the nonrelativistic limit of the Hamiltonian (5) in the plane wave approximation: The Pauli matrixes 1 τ and 2 τ ( 1 τ + , 1 τ − ) are well known: The approximated Hamiltonian (11) is used to describe the nuclear processes with the dineutroneum.
First, we take into account, that mass of dineutroneum is less, than double mass of the neutron. Therefore, neutrino in the atom of dineutroneum is in the bound state, and the Hamiltonian looks like According to a "golden Fermi's rule", the probability of the transition to the continuum states per unit of time is equal: Hence, the decay probability of the bound state of two neutrons and one neutrino per the time unit is equal: Let us now consider the β -decay of the dineutroneum. The initial and final states in this case are 2 : Consequently, the matrix element in (15) We consider the dineutroneum β -decay in its rest system. In this case 0 D k ν = , and (18) is simplified (details see in the Appendix): In turn, eq. (15) can be presented in the form which is suitable for numerical and evaluate the integral All the particles in our case are non-relativistic. Consequently, As a result, and we get the following expression: (32) Only Hulten's WF ( ) with the normalization constant Here We assume that [ ] with [ ] and equal parameters µ for deuteron and dineutroneum.
For the sake of simplicity we suppose According to (30) This integral in a view of (33), (35) and (37) can be calculated analytically where ( ) ( ) The Fermi function ( ) F η in the "point-like deuteron" approximation is equal [ where 2 e ν ν is the probability for the neutrino to be in the state e ν in the dineutroneum.
In the table 1    ♦ without, and ■ with the Fermi factor ( ) F η .  Let us consider the dependence of the dineutroneum lifetime on its size. This dependence should be taking into account, since the triplet length of the neutronneutron scattering much exceeds the deuteron's effective radius d r . Table 2 Table 2 The dependence of rate of the β -decay of the dineutroneum on the ratio 2 / d n α α .
It follows from Table 2, that if the size of dineutroneum alike the size of deuterium mesoatom, its lifetime would be almost 3 seconds. Consequently, one can conclude that the exotic dineutroneum atom is metastable and its lifetime Thus, there is no neutrons' emission, but a creation of helium-3, helium-4 and tritium, and changed natural abundance ratio of palladium isotopes, and a plenty of excess energy. These effects were observed in the experiments [9,10] (see Fig. 3).