Electromagnetic properties of neutrinos: three new phenomena in neutrino spin oscillations

In studies of neutrino electromagnetic properties we discuss three very interesting aspects related to neutrino spin oscillations. First we consider neutrino mixing and oscillations in the mass and flavour bases under the influence of a constant magnetic field with nonzero transversal and longitudinal components. Then we discuss the effect of neutrino spin oscillations induced by electroweak interactions of neutrino with moving matter in case there is matter transversal current or polarization. In the final part of the paper we discuss recently developed approach to description of neutrino spin and spin-flavour oscillations in a constant magnetic field that is based on the use of the exact neutrino stationary states in the magnetic field.


Introduction
α, β = 1, 2 It is well known that massive neutrinos have nontrivial electromagnetic properties, and at least the magnetic moment is not zero [1]. Thus, neutrinos do participate also in the electromagnetic interaction (see [2] for a review). The best terrestrial laboratory upper bound on neutrino magnetic moments is obtained by the GEMMA reactor neutrino experiment [3]. The best astrophysical upper bound was derived from considering stars cooling [4]. The neutrino magnetic moment procession in the transversal magnetic field B ⊥ was first considered in [5], then spin-flavor precession in vacuum was discussed in [6], the importance of the matter effect was emphasized in [7]. The effect of resonant amplification of neutrino spin oscillations in B ⊥ in the presence of matter was proposed in [8,9], the impact of the longitudinal magnetic field B || was discussed in [10].
Here below we discuss three very interesting aspects related to the neutrino spin and spin-flavour oscillations: 1) we consider in details [11] neutrino mixing and oscillations in arbitrary constant magnetic field that have B ⊥ and B || nonzero components in mass and flavour bases , 2) we show that neutrino spin and spin-flavour oscillations can be induced not only by the neutrino interaction with a magnetic field but also by neutrino interactions with matter in the case when there is a transversal matter current or matter polarization [12,13] (see [14] for historical notes and further references). 1 (10) where γ αα ′ = 1 2 m α E α + m α ′ E α ′ is the transition gamma-factor. Introducing an angle β between B and p α vectors and assuming that B ⊥ is aligned along the x-axis we further obtain: As it was expected, in neutrino transitions without change of helicity only the B = B cos β component of the magnetic field contribute to the effective potential, whereas in transitions with change of the neutrino helicity the transversal component B ⊥ = B sin β matters. Performing remaining some simple algebra one can readily write out the H B matrix. For the effective Hamiltonian with the diagonal vacuum part H vac we get This equation governs all possible oscillations of the four neutrino mass states determined by the masses m 1 and m 2 and helicities s = 1 and s = −1 in the presence of a magnetic field. Thus, it follows that: 1) the change of helicity is due to the magnetic (or transition) moment interaction with B ⊥ , 2) the longitudinal field B || , coupled to the magnetic moment, shifts the neutrino energy, 3) an additional mixing between neutrino states with different masses is induced by the magnetic moment interaction with B || .

Oscillations in flavour basis
Once having physics in the mass basis in hands, our next step is to bring it to observational terms [11]. This means that we must elaborate a generalization of the mixing matrix for transitions between neutrino vector written in two four-component bases ν m and ν f = (ν R e , ν L e , ν R µ , ν L µ ) τ so that This procedure appears to be not quite direct since we should hold the condition that the polarization of the fields must preserve under transformation of the bases elements. That is why we put (still keeping in mind that chiral components are almost helicity ones): ν R,L e = ν 1,s=±1 cos θ + ν 2,s=±1 sin θ, ν R,L µ = −ν 1,s=±1 sin θ + ν 2,s=±1 cos θ.
It is interesting to consider a particular case when only the longitudinal magnetic field B || is present in an environment (B ⊥ = 0). Then the Hamiltonian (18) is reduced to the following one, Obviously, the neutrino states with different flavor and same chirality decouple and form subsystems independently mixed by the magnetic field. For example, one would have two states (ν L e , ν L µ ) mixed in accordance with the equation From this it follows that the neutrino magnetic moment interactions with the longitudinal magnetic field can generate the neutrino flavour mixing (an additional mixing to the usual effect due to neutrino mixing angle θ) without changing neutrino chirality. For the flavour neutrino oscillation probability in the adiabtic case we get where It follows that B || not only generates flavour neutrino mixing but also can produce the resonance amplification of the corresponding oscillations.

Neutrino spin precession and oscillations due to matter transversal motion
Consider, as an example, an electron neutrino spin procession in the case when neutrinos with the Standard Model interaction are propagating through moving and polarized matter composed of electrons (electron gas) in the presence of an electromagnetic field given by the electromagnetic-field tensor F µν = (E, B). As discussed in [12,13] (see also [16][17][18]), the the generalized Bargmann-Michel-Telegdi equation describes the evolution of the three-dimensional neutrino spin vector S , where the magnetic field B 0 in the neutrino rest frame is determined by the transversal and longitudinal (with respect to the neutrino motion) magnetic and electric field components in the laboratory frame, The matter term M 0 in Eq. (25) is also composed of the transversal M 0 and longitudinal M 0 ⊥ parts, Here n 0 = n e 1 − v 2 e is the invariant number density of matter given in the reference frame for which the total speed of matter is zero. The vectors v e , and ζ e (0 ≤ |ζ e | 2 ≤ 1) denote, respectively, the speed of the reference frame in which the mean momentum of matter (electrons) is zero, and the mean value of the polarization vector of the background electrons in the above mentioned reference frame. The coefficients ρ (1,2) e calculated within the extended Standard Model supplied with S U(2)-singlet righthanded neutrino ν R are respectively, . For neutrino evolution between two neutrino states ν L e ⇔ ν R e in presence of the magnetic field and moving matter we get the following equation Thus, the probability of the neutrino spin oscillations in the adiabatic approximation is given by (see [12,13]) where E eff = µ B ⊥ + 1 γ M 0⊥ , ∆ eff = µ γ M 0 + B 0 . It follows that even without presence of an electromagnetic field, B ⊥ = B 0 = 0, neutrino spin oscillations can be induced in the presence of matter when the transverse matter term M 0⊥ is not zero. This possibility is realized in the case when the transverse component of the background matter velocity or its transverse polarization is not zero. It is obvious that for neutrinos with nonzero transition magnetic moments a similar effect of spin-flavour oscillations exists under the same background conditions. A possibility of neutrino spin procession and oscillations induced by the transversal matter current or polarization was first discussed in [12,13]. The existence of this effect has been recently confirmed in [19][20][21] where neutrinos propagation in anisotropic media is studied.

Neutrino spin oscillations and stationary spin states in magnetic field
We develop a new approach [15], that is more precise than the usual one, to description of neutrino spin and spin-flavor oscillations in the presence of an arbitrary magnetic field. The derived probability of neutrino oscillations does not coincide with the usual one and the difference might have important phenomenological consequences. Within this customary approach the helicity operator is used for classification of a neutrino spin states in a magnetic field. However, the helicity operator does not commute with the neutrino evolution Hamiltonian in a magnetic field. This case resembles situation of the flavour neutrino oscillations in the nonadiabatic case when the neutrino mass states are not stationary. The proposed alternative approach to neutrino spin oscillations is based on the exact solutions of the corresponding Dirac equation for a massive neutrino wave function in the presence of a magnetic field that stipulates the description of the neutrino spin states with the corresponding spin operator that commutes with the neutrino dynamic Hamiltonian in the magnetic field.
Here we again consider a simple model with two generations of flavour neutrinos ν e and ν µ that are the orthogonal superpositions of mass states ν 1 and ν 2 ν f = i U f i ν i , where U f i are elements the mixing matrix given by (15) and f = e, µ, i = 1, 2. We start with consideration of a massive ν i with the magnetic moment µ i that propagates along n z direction in presence of constant homogeneous arbitrary orientated magnetic field B = (B ⊥ , 0, B ). The neutrino wave function in the momentum representation is given by a plane wave solution of the modified Dirac equation The neutrino energy spectrum can be determined from the condition which guarantees the existence of a nontrivial solution of the modified Dirac equation (32). For the neutrino energy spectrum we obtain where "±" denotes two different eigenvalues of the Hamiltonian H i = γ 0 (m i + γp + µ i σB), which describes dynamics of the neutrino system under consideration. We define different neutrino spin states in the mass basis as eigenstates of the spin operator which commutes with the Hamiltonian H i . Hence, we specify the neutrino spin states as the stationary states for the Hamiltonian, contrary to the case when the helicity operator is used. Consider the mass state ν i as a superposition of neutrinos ν + i and ν − i in a definite spin state, The complex coefficients denote two different eigenstates of the spin operator S i and |c + i | 2 + |c − i | 2 = 1. Thus, the neutrino mass states evolve following to where the neutrino initial state at t = 0 is given by ν ± i (t = 0) = ξ ± i e i px . In the following calculations the term e i px is neglected because it is irrelevant for the neutrino oscillation probability.
Next we assume that the initial neutrino state ν(t = 0) is a pure electron state which is defined as the superposition of the mass states, Using (36) we see that this state depends on time as Therefore, the probability to observe the muon neutrino state ν µ at time t is given by P ν e →ν µ (t) = (37) and (38) we get for the oscillation probability It is usually assumed that the initial state of relativistic neutrino is a negative-helicity state, which means that σp |p| Next we consider the left-handed spinors because only the left-handed fermions participate in the production and detection processes and we suppose that each of the mass states of the initial electron neutrino are left-handed. In our case the helicity operator is equal to σp/|p| = σ 3 , therefore the initial neutrino state is given by ψ L = (0, 1, 0, 0) T . Let us write the initial neutrino state ψ L as a superposition of the eigenvectors of the spin operator S i . From (35) we get where φ is the angle between B and p. It is obvious that S 2 i = 1 + p 2 m 2 i sin 2 φ Î 4×4 . In order to define the spin projector operators we introduce the normalized spin operator following tõ The spin projector operators are P ± i = 1±S i 2 , and we use them to split the initial neutrino state ψ L in two neutrino states with definite spin quantum numbers Note that where η ± i is a basis in the spin operator S i eigenspace. From the condition ψ †± i ψ ± L = |c ± i | 2 we get that |c + i | 2 = 1−N i cos φ 2 , |c − i | 2 = 1+N i cos φ 2 . Now we can insert the obtained expressions for |c ± i | 2 in (39). In the forthcoming evaluation of the probability P ν e →ν µ (t) we consider the case when the magnetic field B is nearly a transversal one and B ⊥ ≫ B , therefore sin φ ≈ 1, cos φ ≈ 0. Then we get (it is also supposed that p 2 m 2 i ≫ 1)