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On the Electric Charge of the Neutrino


ON THE ELECTRIC CHARGE OF THE NEUTRINO

Rasulkhozha S. Shara?ddinov

Institute of Nuclear Physics, Uzbekistan Academy of Sciences, Tashkent, 702132 Ulugbek, Uzbekistan

arXiv:hep-ph/0307083v1 7 Jul 2003

Exact expression is obtained for the di?erential cross section of elastic electroweak scattering of longitudinal polarized massive Dirac neutrinos with the electric charge and anomalous magnetic moment on a spinless nucleus. This formula contains all necessary information about the nature of the neutrino mass, charge and magnetic moment. Some of them state that between the mass of the neutrino its electric charge there exists interconnection.

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From time to time in leterature the considerations in favor of the existence of a small electric charge eν , magnetic moment ?ν or electromagnetic radius rν of the neutrino with a non - zero rest mass mν are listed and some possible consequences of their interaction with other particles [1, 2, 3, 4] are considered. Reactions with electrons and nucleons, and also the neutron and muon decays apply to them. Exactly the same one can as the source of the information choose the processes on the nuclear targets. In the case of a nucleus with the electric (Z ) and weak (ZW ) charges on the elastic scattering [5] ν (ν ) + A(Z, ZW ) → ν ′(ν ′) + A(Z, ZW ),
γ,Z 0

(1)

in?uence only the properties of the neutrino itself. In this connection appears of principle possibility to rewatch the behavior of the neutrino in the elastic scattering on a spinless nucleus. Such an analysis is, in particular, carried and out in the present work. It is assumed that the neutrino has the longitudinal polarization. The scattering amplitude at the account of hadronic and leptonic weak neutral currents may to the lower order in α and GF be written as
ew em we Mf i = Mf i + Mf i =

GF Z0 = √ u(p′, s′ )γ?(gVν + γ5gAν )u(p, s)J? (q )+ 2 4πα γ + 2 u(p′, s′)[γ?F1ν (q 2 ) ? σ?λqλ F2ν (q 2)]u(p, s)J? (q ), (2) q where p(p′) and s(s′ ) denote the four - momentum and helicity of initial (?nal) neutrinos, q = p ? p′ , γ? and σ?λ = (γ? γλ ? γλ γ? /2 are the Dirac matrices, x gVν and gAν characterize the vector and axial vector coupling constants, J? (q ) imply the nucleus electromagnetic (x = γ ) and weak neutral (x = Z 0) currents, F1ν (q 2) and F2ν (q 2 ) describe the neutrino Dirac and Pauli form factors which are in the static limit (q 2 = 0) reduced to the values F1ν (0) = eν , F2ν (0) = ?ν . In the case of a zero - spin nucleus, the di?erential cross section of the process (1) on the basis of (2) one can present as dσew = dσem + dσint + dσwe.
2

(3)

First term here answers to purely electromagnetic scattering and equal to dσem 1 ν 2 ?1 2 = σo (1 ? ην ) {(1 + ss′ )F1 ν+ d? 2 θ 2 2 2 ?2 +ην (1 ? ss′ )[F1ν ? 2mν (1 ? ην )F2ν ]2tg 2 }FE (q ). (4) 2 The contribution explained by the interference of neutral and electromagnetic currents has the form 1 ν gA dσint 2 ?1 2? F + = ρσo (1 ? ην ) gVν {(1 + ss′ ) ?1 ? s ν 1 ? ην 1ν d? 2 gVν θ 2 ?2 +ην (1 ? ss′ )[F1ν ? 2mν (1 ? ην )F2ν ]tg 2 }FEV (q 2). 2 One can write also the cross section of purely weak transition
2 2 Eν GF θ dσwe 2 2 = {(gV + gA )(1 + ss′ )cos2 + ν ν 2 d? 16π 2 ? ?

(5)

θ 2 2 2 ′ 2θ +ην [gVν (1 ? ss′ )sin2 ? gA (1 + ss ) cos ]? ν 2 2 θ 2 cos2 }F 2 (q 2 ), (6) ?2sgVν gAν (1 + ss′ ) 1 ? ην 2 W where the upper (lower) sign corresponds to the neutrino (antineutrino). Here one must have in view of that
ν σo θ α2 cos2 2 GF q 2 mν = , ρ= √ , , ην = 2 (1 ? η 2 )sin4 θ E 4Eν 2π 2α ν ν 2

FE (q 2 ) = ZFc(q 2), FEV (q 2) = ZZW Fc2 (q 2), FW (q 2) = ZW Fc (q 2 ), 1 (0) (1) ZW = {βV (Z + N ) + βV (Z ? N )}, 2 from which Eν is the neutrino energy, Fc is the charge (Fc(0) = 1) contribu(1) (0) tion form factor of a nucleus with neutrons number N = A ? Z, βV and βV are the isoscalar and isovector constants of vector neutral hadronic current. The terms depending upon (1 + ss′ ) and (1 ? ss′ ) imply that only the conservation (s′ = s) or only the change (s′ = ?s) of the neutrino helicity is responsible for the scattering. As seen from (4), if the neutrino has the
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magnetic moment F2ν (q 2 ) then the left - handed (s = ?1) neutrino in the reaction (1) can be converted into the right - handed (s = +1) one, and vice versa [5]. It is clear that interconversions νL ? νR , ν R ? νL (7)

are carried out in the nucleus ?eld both in reusulting the in?uence of the electric charge F1ν (q 2) on the neutrino polarization and in the weak interaction of slow leptons with hadrons. In the latter case from (6), we are led to the consequence of the Dirac equation that the neutrino must change his helicity at the availability of a non - zero rest mass [6]. On the other hand, it is known that in the framework of the (V ? ) version of the theory, between ?ν and mν there exists a sharp dependence [7] owing to which, the left - and right - handed neutrinos magnetic moments answer to the ?ip of their spins. According to such a point of view, the processes of interconversions (7) originated in the nucleus ?eld at the expense of form factor F1ν (q 2 ) re?ect the availability of an intimate connection between the mass of the neutrino and its electric charge.

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References
[1] J. Bernstein, M. Ruderman and G. Feinberg, Phys. Rev. 132 (1963) 1227 [2] J. Panman, in Proc. Int. Symp. on Lepton and Photon Interactions at High Energies, Hamburg, 1987 (North - Holland, Amsterdam, 1987), p.553 [3] S. Davidson, B. Campbell and K.D. Bailey, Phys. Rev. D 43 (1991) 2314 [4] I.S. Batkin and M.K. Sundaresan, J. Phys. G 20 (1994) 1749 [5] R.S.Shara?ddinov, Institute of Nuclear Physics, Uzbekistan Academy of Sciences, preprint No. P-12-325 (1988) 10pp. [6] Ya.B. Zel’dovich and M.Yu. Khlopov, Uspehi Fiz. Nauk. 135 (1981) 45 [7] K. Fujikawa and R.E. Shrock, Phys. Rev. Lett. 45 (1980) 963

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