Neutrino-bound di-neutrons as an exotic metastable atom Yu.L. Ratis Institute of power engineering for the special application 443071, Volgsky pr.33-87, Samara, Russia, [email protected] Image processing systems institute RAS 443001, Molodogvardeiskaya, 151, Samara, Russia, [email protected] Abstract 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 re- action of deuterons with free or quasi-free electrons. The estimation of mass, size and lifetime of dineutroneum atom is fulfilled. 1. 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]. Fig. 1. The typical diagram of the electroweak process [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 W - boson 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 neu- tron) 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 po- tentials 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 1 this in more detail. If the energy of incoming electron is resonant (i.e. renormalized masses of all three types of neutrinos (νννe,,µ τ ) 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 Dν is metastable, because the energy conservation law forbids its decay with µ - or τ - lepton emission. The channel Dnν → 2 + νe is also closed. Thus, theoretical consideration of the bound state of the neutrino inside a nucleus in the framework of any potential model gives us only phenomenological description of the observable effect. 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. 2. Main formalism The known Hamiltonian of weak interaction is G GGGGGG H′′′′= Jλ+() r⋅⋅ Gˆ (, r r ) J ( r ) drdr , (1) 2 ∫ λ G with G constant of universal weak interaction Fermi, Jrλ() weak current, and GG Grrˆ(,′ ) propagator. Let us introduce definition in accord to [6] ⎪⎧ λ+ + ⎪JJ==( λ ) ,1,2,3λ ⎨⎪ , (2) ⎪JJ4+ = −()+ ⎩⎪ 4 and similarly for others 4- vector operators. In the standard model, the weak inter- action is caused by exchange of the W - boson with mass ~90GeV . Therefore, if we consider the low energy weak processes, an approximation mW →∞can be used. Accordingly, the interaction is quite local, and components of the weak cur- rent in Hamiltonian (1) should be taken at the same point of space G GGG (Grrˆ(,′′ )= δ ( r− r )). Hence 2 G GGG HJrJrdr′ = λ+()⋅ () . (3) 2 ∫ λ The Lorenz invariant weak current is well known. For example, β - decay of a neutron is described by the Hamiltonian [6] G GGGGG+ Hr′ =+⎡⎤ψγ()λ (1 γψ ) () rrrdr⋅ ⎡ ψγ () (1+ γψ ) ()⎤ . (4) 2 ∫ ⎣⎦npe55⎣⎢ λνe ⎦⎥ To describe the weak processes in nuclear physics, one needs in a non-relativistic G Hamiltonian hr′(). The model of the Hamiltonian was derived in the early papers of Fermi, Gamov and Teller, and looks like [6] GGG + λ hrt′(,)=++++{} iβγ [ f12λλρρλ f σ k ( g 1 γ igk 25 λ ) γ ] j (,) rt hc .. (5) 2 In (5) G GGi λλ⎡⎤⎛⎞⎟ jrt(,)=+ iψγll () r() 1 γψ5 νν () r⋅− exp⎜ ( E − Et )⎟ (6) ⎣⎦⎢⎥ll⎝⎠⎜ = ⎟ the lepton current, E the energy positive for particles and negative for antiparti- G cles, f1 , f2 , g1 , g2 the formfactors, ψ()r lepton wave function (WF). In the works devoted to the nuclear β - processes, the WFs of free leptons in G 1 (6) are usually chosen as plane waves with the momentum pl . Thus, the lepton’s current (6) looks like: G G G i −3 ⎛⎞⎟ jrtλλ(,)= Lb exp( ikr⋅⋅ )exp⎜ − ( E ν − Ete )⎟ (7) ⎝⎠⎜ = ⎟ G G G G where ke= ν − the lepton transferred momentum, ν the wave vector of neutrino, G e the wave vector of electron, L3 the normalization volume, bmmλν(,eee )= ium( ()γ λνν wm ()) , (8) and wmνν()(1)()=+γ5 um νν. (9) The spinor 1 ⎛⎞1 GG ⎜ ⎟ ˆ wmνν()= ⎜ ⎟⋅−() 1(σν ⋅ ) ⋅ χ1/2 () m ν , (10) 2 ⎝⎠⎜−1⎟ 1 In reactions of electron capture, β - decay into a bound state and in mesoatoms the charged lepton occupies the bound state and its WF belongs to the discrete spectrum. 3 mν =±1/2 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 =kMcpMc/( ), / и kR due to the forbidden transitions, and obtain the non- relativistic limit of the Hamiltonian (5) in the plane wave approximation: G G G G A G GGG ′ ik⋅ r ⎡⎤ hr( )= 3 e⋅⋅−⋅⋅⋅−∑ ⎢⎥ ifb14 g 1( bστδ) (+ )jj ( r r )+ ... (11) 2 ⋅ L j=1 ⎣⎦j The Pauli matrixes τ1 and τ2 ( τ+1 , τ−1 ) are well known: ⎪⎧τττ=+()/2/2ipnp =− τ⎧ τ== 0; τ ⎪ ++++12 1 ⎪ ⎨⎨⎪⎪⇒ . (12) ⎪⎪τττ= ()/2/2− i = τ ττnpn==0; ⎩⎪⎪⎪−−12 1 ⎩⎪ −− 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 GGGGGG GG ⎪⎪⎧⎫2 G hr′()= β ψδλστ ( r )e⋅⋅−⋅ie r⎨⎬⎪⎪ ( r − r )⎡⎤ ib −⋅⋅ ( b() i ) () i + hc .., (13) 3/2 ν ci⎪⎪∑ ⎣⎦⎢⎥4 + 2 ⋅ L ⎩⎭⎪⎪i=1 G ψν ()rc the spatial part of the neutrino’s WF, GfGβ = 1 ⋅ , index c indicates the ra- dius-vector of the neutrino which origin is in the centre- mass of the dineutroneum G because of translation-invariance of the Hamiltonian hr′(). According to a “golden Fermi’s rule”, the probability of the transition to the continuum states per unit of time is equal: 2π 2 dw= ⋅−⋅δ () E E f V i dn . (14) fifif= Hence, the decay probability of the bound state of two neutrons and one neutrino − within the channel Ddeν → + per the time unit is equal: 33G G 2 2π Ldped Ldp G ()N G wEEdhrDdr− = ⋅⋅−⋅δ()if ′′ ()ν ′ . (15) Ddeν → + ===∫∫(2ππ )33 (2 ) 4 ()N The WFs Dν and d depend on the coordinates, spins and isospins of nucleons, − and matrix elements of the transition Ddeν → + in the space of leptons are al- G ready included into the Hamiltonian hr′() by definition. The external triangular brackets in (15) mean the averaging by projections of spins of all initial particles, and analogous summation in the final state. Let us now consider the β - decay of the dineutroneum. The initial and final states in this case are2: G G G G ⎪⎧ ()N 1 ikDD R G G ⎪ DerrST= ννψχχ()()()− ⎪ ν 3 22n 10011− ⎪ L ⎨⎪ (16) ⎪ 1 GG GG G G ⎪ derrST= ikdd R ψχχ()()()− ⎪ dm211d 00 ⎩⎪ L3 Consequently, the matrix element in (15) looks like G G dh′′() r D()N dr ′= ∫ ν GG GG GGG G (17) 1 GGGik( DD R− kR dd) G G G = dr′′′′′′′ drdr eνν ψψ*() r () r χ++ ()()() S χ T h r χχ () S () T L3 ∫ 12dn 2 1 md 00 00 11− G GG where rrr′′ = 21− . The "nuclear" spin of the dineutroneum J i = 0 and the deuteron’s spin J f = 1 . Thus, we deal with the Gamov - Teller transition. According to it G −⋅λ G GGGGGG ⎪⎪⎧⎫2 G hr′ ()= β ψδστ ()e r⋅⋅−⋅ie r⎨⎬⎪⎪ ( rr −⋅⋅⋅ )( b() i ) () i + hc .. (18) GT3/2 ν c⎪⎪∑ i + 2 ⋅ L ⎩⎭⎪⎪i=1 We consider the dineutroneum β - decay in its rest system. In this case k = 0 , and (18) is simplified (details see in the Appendix): Dν G GG 2 G G G GGλ ⋅Gβ 3 1/2me GGGG dh′′( r ) D()NikRier dr ′= C dRdre−−⋅diψψ * () r () r ψ ( r−⋅ R )e (19) ∫∫ν 9/2 11/2−mmd ν d 2 n∑ ν i 2L i=1 We determine the formfactor ()N GGGGGGG−1/2 fedD⇔ ν ( )= cos( errrrdrV⋅≡ /2)ψψ* ( ) ( /2) ψ ( ) Dν overlap∫ dν 2 n() eff . (20) Dν The Veff means an effective volume of exotic atom of dineutroneum. This circum- stance allows to present eq. (19) in the extremely compact form: 2 See details in [17]. 5 G ()N GGλ ⋅Gβ 2 GG 1md dh′′() r D()NdDm dr ′=+ (2)(πδ 3 k e ) f⇔ νν ()(1) e− 1/2+ C . (21) ∫ ν L9/2 d overlap1/2− me 1/2 mν In turn, eq. (15) can be presented in the form which is suitable for numerical calculations G 2π dp ()N G 2 e dD⇔ ν wEEGfe− = ⋅−⋅⋅δλ()3()i fβ overlap , (22) Ddeν → + ==∫ (2π )3 and evaluate the integral ph G 2 I− () peeifeeDde= dp⋅−δπδ ( E E )= 4 dp p ( E−− E E ) (23) Ddeν → + ∫∫ν All the particles in our case are non-relativistic.