Spin-flavor oscillations of Dirac neutrinos in a plane electromagnetic wave

We study spin and spin-flavor oscillations of Dirac neutrinos in a plane electromagnetic wave with circular polarization. The evolution of massive neutrinos with nonzero magnetic moments in the field of an electromagnetic wave is based on the exact solution of the Dirac-Pauli equation. We formulate the initial condition problem to describe spin-flavor oscillations in an electromagnetic wave. The transition probabilities for spin and spin-flavor oscillations are obtained. In case of spin-flavor oscillations, we analyze the transition and survival probabilities for different neutrino magnetic moments and various channels of neutrino oscillations. As an application of the obtained results, we study the possibility of existence of $\nu_{e\mathrm{L}}\to\nu_{\mu\mathrm{R}}$ oscillations in an electromagnetic wave emitted by a highly magnetized neutron star. Our results are compared with findings of other authors.


Introduction
The experimental confirmation of oscillations of atmospheric and solar neutrinos [1,2] is the direct indication that neutrinos have nonzero masses and mixing, which, in its turn, unambiguously points out to the physics beyond the standard model.This experimental success was followed by the determination of other parameters in the neutrino mixing matrix, including θ 13 (see, e.g., ref. [3]) and the CP violation phase (see, e.g., ref. [4]).
Despite the great importance of neutrino flavor oscillations for the experimental studies of properties of these particles, other channels of neutrino oscillations are of interest for the evolution of astrophysical and cosmological neutrinos.In the present work, we shall deal mainly with spin-flavor neutrino oscillations, which imply the conversion of the type ν βL → ν αR , where both flavor, α, β = e, µ, τ, . . ., and helicity, L, R, change.This type of neutrino transitions implies that these particles possess nonzero magnetic moments and interact with a strong electromagnetic field.Neutrino electromagnetic properties are reviewed in ref. [5].
The majority of studies of neutrino spin-flavor oscillations involve the neutrino interaction with a constant transverse magnetic field.Other configurations of electromagnetic fields, including an electromagnetic wave, were used in refs.[6,7] in the examination of spin-flavor oscillations.The neutrino interaction with an electromagnetic wave is important for the studies of neutrino propagation in strong laser pulses [8,9].
In the present work, we study neutrino spin and spin-flavor oscillations in a plane electromagnetic wave on the basis of the exact solution of the Dirac-Pauli equation for a massive neutrino in this external electromagnetic field [10].In our analysis, we suggest that neutrinos are Dirac particles.Despite some theoretical models for the neutrino mass generation point out that neutrinos are likely to be Majorana particles [11], the issue of the neutrino nature is still open [12].
Our work is organized as follows.In section 2, we start with the studies of neutrino spin oscillations in a plane electromagnetic wave with circular polarization within one neutrino mass eigenstate.Then, in section 3, we generalize our treatment to account for neutrino spinflavor oscillations with the great transition magnetic moment.The description of spin-flavor oscillations is based on the formulation of the initial condition problem for flavor neutrinos [13].An astrophysical application is also considered in section 3. The influence of the diagonal magnetic on spin-flavor oscillations is studied in section 4. We summarize our results in section 5.The matrix elements of the neutrino spin interaction are calculated in appendix A.

Spin oscillations in an electromagnetic wave
In this section, we consider neutrino spin oscillations, within one neutrino generation, in a plane electromagnetic wave with the circular polarization.Our analysis is based on the exact solution of the Dirac-Pauli equation for a massive neutral fermion found in ref. [10].
We assume that a neutrino is a Dirac particle.In this section, we shall neglect the mixing between different mass eigenstates.Assuming that the considered neutrino mass eigenstate has the nonzero mass m and the magnetic moment µ, the Dirac equation for such a neutrino, described by the bispinor ψ, in the external electromagnetic field where γ µ = (γ 0 , γ) and σ µν = i 2 [γ µ , γ ν ] − are the Dirac matrices.In the following, we shall use the standard representation for the Dirac matrices [14].
We shall take that the external electromagnetic field is in the form of a plane electromagnetic wave propagating in the positive direction of the z-axis.It is convenient to choose the following gauge for the vector potential: A µ = (0, A).Neglecting the dispersion of the wave, we can take that A = A(t − z).The electric and magnetic fields are E = −A ′ and B = (e z × E), where the prime means the derivative with respect to the whole argument of A.
The exact solution of eq.(2.1) was found in ref. [10] for the arbitrary propagation of the fermion with respect to the electromagnetic wave.In the present work, we consider a special situation when a neutrino propagates along the wave, i.e. ψ = ψ(z, t).In this case, the solution of eq.(2.1) has the form, where v = v(t − z) is the two component spinor, E = p 2 + m 2 is the neutrino energy, p is the neutrino momentum, and σ = (σ 1 , σ 2 , σ 3 ) are the Pauli matrices.It is interesting to notice that The spinor v obeys the equation [10], To proceed with the analysis of the solution of eq. ( 2.3) we suppose that we have a circularly polarized electromagnetic wave, i.e.
and B z = 0, where B 0 is the amplitude of the wave and ω is its frequency.In this case, the solution of eq. ( 2.3) has the form, where φ = t − z is the current phase, φ 0 = t 0 − z 0 is the initial phase, v 0 = v(φ 0 ) is the initial spinor, and Ω = (µB 0 Now we should specify the initial condition.Let us suppose that v T 0 = (0, 1).In this case, using eq.( 2.2), one can check that P − ψ 0 = ψ 0 , where ψ 0 is the initial bispinor, corresponding to v = v 0 , P ± = (1 ± Σ z )/2 are the helicity projection operators, Σ = γ 5 γ 0 γ and γ 5 = iγ 0 γ 1 γ 2 γ 3 are the Dirac matrices.It means that initially a neutrino is left-handed, i.e. its spin is opposite to the particle momentum.
Neutrino spin oscillations correspond to the appearance of the nonzero component ψ + , i.e. the right-handed polarization, at φ > φ 0 , where ψ + = P + ψ.The probability of L → R transitions is P L→R = ψ † P + ψ.Using eq.(2.2), one can show that P L→R = 1 + v † σ 3 v /2, i.e. the transition probability is completely defined by the evolution of the neutrino spin in eq.(2.3).
After lengthy but straightforward calculations, using eq.(2.4), we can represent P L→R in the form, One can see in eq.(2.5) that the transition probability is a function of ∆φ = φ − φ 0 , i.e. it depends on both t and z.Thus neutrino spin oscillations can happen both in time and in space.It is owing to the fact that the spinor v in eqs.(2.2) and (2.4) depends on φ.It is the feature of the field theory based approach to the description of neutrino oscillations.
The coordinate dependence of P L→R in eq.(2.5) becomes important if a neutrino interacts with an electromagnetic wave for a quite long time.In such a situation, the particle initial wave packet becomes broad enough.It can happen for relatively low energy neutrinos.In practice, a neutrino is an ultrarelativistic particle.We can localize it and attribute a mean velocity β = p/E of its wave packet.Thus z = βt and z 0 = βt 0 .
Using this way to modify the coordinate dependence of P L→R in eq.(2.5), we rederive the transition probability of neutrino spin oscillations in a circularly polarized electromagnetic wave obtained in ref. [6].For this purpose one should set ∆m 2 = 0 and neglect the interaction with matter in the corresponding expressions in ref. [6] since, in this section, we study neutrino spin oscillations induced by the electromagnetic wave only within one neutrino generation.In the following section, we shall generalize this formalism to describe neutrino spin-flavor oscillations.
3 Spin-flavor oscillations: great transition magnetic moment In this section, we shall extend the formalism developed in section 2 to describe neutrino spin-flavor oscillations.For this purpose we discuss the system of two flavor neutrinos and formulate the initial condition problem for these particles.Here we will focus on the case of the great transition magnetic moment.
For simplicity, we shall consider the system of two flavor neutrinos (ν α , ν β ).The generalization of the formalism for greater number of neutrino flavors is straightforward.For instance, we can define ν β = ν e and ν α = ν µ,τ .The Lagrangian of this system in the presence of the external electromagnetic field reads where (m λλ ′ ) and (M λλ ′ ) are the matrices of masses and magnetic moments in the flavor eigenstates basis.As in section 2, we shall suppose that only an electromagnetic wave, propagating along the z-axis, is present in the system and neutrinos move in the direction of the wave.We shall define the initial conditions in the system described by the Lagrangian in eq.(3.1).For this purpose, we shall suppose that, at t = 0, ν α = 0 and ν β (z) = f ν iL , where f = f (z) is the given coordinate dependence of the initial wave function and ν iL is a constant bispinor.Let us suppose that Σ z ν iL = −ν iL .For example, we can take ν T iL = (1/ √ 2)(0, 1, 0, −1).Such a choice of the initial wave functions corresponds to the presence of left-handed electron neutrinos (if ν β ≡ ν e ) and the absence of neutrinos of other flavors.
If only two neutrino flavors are present in the system, we can introduce the neutrino mass eigenstates ψ a , a = 1, 2, to diagonalize the mass matrix in eq.(3.1).The corresponding matrix transformation reads where θ is the vacuum mixing angle.Note that we can say whether neutrinos are Dirac or Majorana only after the transition to the mass eigenstates basis in eq.(3.2).The Dirac equation for ψ a has the form, where Here µ a ≡ (µ 11 , µ 22 ) and µ ≡ µ 12 are the diagonal and transition magnetic moments in the mass eigenstates basis, (µ ab ) = U † M U , as well as α = γ 0 γ and β = γ 0 are the Dirac matrices.We shall discuss the situation when µ a ≪ µ.If we neglect the antiparticle degrees of freedom, the general solution of eq. ( 3.3) has the form, where E a = p 2 + m 2 a are the energies of different mass eigenstates, v T + = (1, 0) corresponds to a right-handed neutrino and v T − = (0, 1) to a left-handed particle since σ 3 v ± = ±v ± .Note that we omit the index a in the spinors v s since we neglect µ a .We also mention the orthogonality of the basis bispinors, u † a± u a∓ = 0.The coefficients a as are supposed to be c-number functions rather than operators acting on Fock states.If we assume that they depend on neither t nor z, the wave functions in eq.(3.5) would satisfy the equations i ψa = H a ψ a .To account for the potential V in eq.(3.4), which mixes different mass eigenstates in eq. ( 3.3), we have to suppose that a as are no longer constant.Such an approach for the description of the evolution of neutrino mass eigenstates in the presence of external fields was used in ref. [13].However, unlike ref. [13], where the corresponding coefficients depend on time only, here we suppose that a as = a as (t − z).Our main goal is to study the behavior of these coefficients.
Substituting the ansatz in eq.(3.5) to eq. ( 3.3), we obtain the equation for a as , Accounting for the mean values we can rewrite eq.(3.6) in the form, In eqs.(3.7) and (3.8), we suppose that one deals with a circularly polarized wave.Before we proceed with the analysis of the evolution of a as we should specify the initial conditions for them.As in section 2, we shall suppose that neutrinos are localized along their trajectories.However, since different mass eigenstates have different masses, we assume that z = βt, where β is the center of inertia velocity [15] Therefore, if t = 0, a as should be taken at zero argument: a as (0).
Accounting for the initial condition for the flavor eigenstates specified above, we obtain that ) and a a+ (0) = 0. Taking into account the explicit form of the initial bispinor ν iL and u a− in eq.(3.5), we get that Taking into account the dependence of a as on t in eq.(3.9), a as = a as (t[1 − β]), we can rewrite eq.(3.9) as the effective Schrödinger equation, where Ψ T = (a 1+ , a 1− , a 2+ , a 2− ) and In eqs.(3.12) and (3.13), we use the approximation of ultrarelativistic neutrinos.One can see that H eff in eq.(3.11) is non-Hermitian since ξ = 1 because m 1 = m 2 .This fact results from the assumption that both mass eigenstates with different masses propagate with the same velocity β.Nevertheless we can approximately set ξ = 1.Indeed we can assume that m a ∼ 1 eV [16].Below, we shall be interested in the ν e ν µ -oscillations channel.In this situation, δm 2 = m 2 1 − m 2 2 ≈ 7.6 × 10 −5 eV 2 [17].Therefore |m 1 − m 2 |/m a ∼ 10 −2 .Hence, we can take that ξ = 1 with a sufficient level of accuracy.Note that we should keep δm 2 = 0 in Φ ± in eq.(3.12) since these terms contribute to the phase of neutrino oscillations, which is very sensitive to the change of parameters.
The solution of eq.(3.14) has the form, where and are the eigenvectors of Heff : Using eqs.(3.10), (3.15) and (3.16), we find that the coefficients a a+ (t) have the form, The explicit form of a a− (t) is not important for our purposes since these coefficients do not contribute to the evolution of the right-handed neutrino states.We are interested in the appearance of right-handed neutrinos of the flavor α in a beam initially consisting of ν βL .We shall suppose that the initial wave packet is quite wide, i.e. we take that f (p) = 2πδ(p − p 0 ), where p 0 is the initial momentum.In the following, we shall omit the subscript 0 for brevity.Using eqs.(3.2), (3.5), and (3.17), we obtain that the wave function ν αR has the form, where
The mentioned discrepancy is because of the unsubstantiated account of the neutrino vacuum oscillations phase δm 2 A(θ)/4p, accompanied by the function of the mixing angle A(θ), in the dynamics of the neutrino spin in ref. [6].In section 2, using the method of the exact solution the Dirac-Pauli equation, we have confirmed that the results of ref. [6] are applicable for the description of the neutrino spin evolution within one mass eigenstate.Thus, in the present work, we have generalized the findings of ref. [6] to treat neutrino spin-flavor oscillations correctly.
To analyze eq. ( 3. 19) we suppose that δΩ ≪ Ω ± , where δΩ = (Ω + − Ω − ) /2.In this situation, the function P ν βL →ν αR becomes rapidly oscillating, being modulated by a slowly varying envelope function.Therefore we can apply the method of the analysis of this envelope function developed in refs.[19,20].Let us represent P ν βL →ν αR in the form, where are the minimal value and the amplitude of the envelope function A eff (t).In eqs.(3.20) and (3.21), Ω 0 = µ 2 B 2 0 (1 − β) 2 + (δm 2 /4p) 2 .The behavior of the transition probability for the ν eL → ν µR oscillations channel for different ω is shown in figure 1.One can see that P ν eL →ν µR is a rapidly oscillating function.Therefore the averaged signal P (t) = A min /2 + δA sin 2 (δΩt) will be detected.The maximal averaged transition probability reads The envelope function A eff (t) and the averaged transition probability P (t) are also shown in figure 1.
The maximal averaged transition probability, calculated using eq.(3.22) with the parameters corresponding to figure 1(a), is Pmax ≈ 0.35.This result is in the agreement with figure 1(a).We can also estimate the the modulation length L = π/δΩ.Using eq.(3.20), one gets that Basing on the parameters corresponding to figure 1(a) and eq.(3.23), we obtain that L ≈ 45 au,1 which again in the agreement with figure 1(a).
As an application of the obtained results, we shall consider ν eL → ν µR spin-flavor oscillations of astrophysical neutrinos in the vicinity of a neutron star (NS).A magnetic field with the strength ∼ 10 18 G, chosen in figure 1, is slightly weaker than the strongest field allowed in NS [22].The frequency ∼ 10 3 s −1 , used in figure 1 frequency of a millisecond pulsar.Therefore an electromagnetic wave with such a characteristics (B 0 = 10 18 G and ω = 10 3 s −1 ) can be emitted by a rapidly rotating highly magnetized NS.
The half of the modulation length, L/2, corresponds to the maximal conversion of the initial neutrino beam.Using the averaged transition probability in figure 1(a), one finds that L/2 ∼ 20 au.This length is comparable with a size of a planetary system.Therefore the described neutrino spin-flavor oscillations may well happen in the vicinity of NS.
The most powerful neutrino emission takes place during a supernova explosion.In several minutes after an explosion, the neutrino flux is rather weak to be detected.Nevertheless, when the NS temperature is above 10 7 K ∼ 1 keV, neutrinos carry away energy from NS and are an effective tool for the NS cooling [23].That is why we take the neutrino energy ∼ 1 keV in figure 1.Thus such neutrinos, produced in modified Urca processes in the NS core, are subject to the described spin-flavor oscillations.Basing on Pmax in figure 1(a), one gets that about 35% of ν eL will be converted to ν µR .
It should be noted that the chosen parameters B 0 and p correspond to δA 1 in eq.(3.21).It guarantees that the transition probability is not suppressed significantly.
Figure 1(b) is unlikely to be implemented in astrophysical environments because of a huge frequency of the electromagnetic wave ω = 10 13 s −1 .It is provided to be compared with figure 1(a) to demonstrate the dependence of P ν eL →ν µR on ω.
At the end of this section, we discuss the general issue on the approximations used in the derivation of the main results.To obtain eq.(3.10) we supposed that neutrino mass eigenstates are localized in the vicinity of their classical trajectories.However, to derive eq.(3.18) one have to assume that the neutrino wave packets are wide.The latter assumption it is necessary for a significant overlap of different mass eigenstates for the oscillations process to occur.Formally we should require that λ ≫ δΩ −1 , where λ is the width of the wave packet.However we can suggest that point-like neutrinos with equal velocities are coherently emitted during the time interval ∆t ≫ δΩ −1 .In this case, one deals with an effectively wide neutrino wave packet consisting of mass eigenstates propagating with a certain velocity.Thus we can reconcile both assumptions necessary to get eqs.(3.10) and (3.18).
4 Spin-flavor oscillations: influence of diagonal magnetic moments In this section, we discuss the influence of the diagonal magnetic moments µ a on the neutrino spin-flavor oscillations in a plane electromagnetic wave.In our analysis, we shall suppose that µ a ≪ µ.
If one accounts for µ a , the main formalism, developed in section 3, remains practically unchanged.First, eq.(3.5) is modified, so that the two component spinor v s , entering to the bispinor u as , becomes dependent on time and µ a , i.e. v s → v as (t).The temporal dependence of v as is given by eq.(2.4).We can choose two independent spin states v 0s in eq.(2.4) as v T 0+ = (1, 0) and v T 0− = (0, 1).In this situation, both v a± (t) and u a± , corresponding to the opposite spin states, are orthogonal.
Second, the matrix elements in eq.(3.8) become quite cumbersome.The calculation of the matrix elements is provided in appendix A in the general form.Using eq.(A.4) and making the calculations similar to those in section 3, we derive the analogue of eq.(3.11), In eqs.(4.1) and (4.2), we set z = βt and assume that m 1 ≈ m 2 in off-diagonal terms ∼ exp(±iΦ ± t) in H eff .Since we discuss the situation when m 1 ≈ m 2 , we should suppose that µ 1 ≈ µ 2 = µ ′ , since µ a ∼ m a ; cf.ref. [24].In this case, ǫ ± = ±ǫ, where ǫ = 4µ ′ B 0 sin 2 ωt(1 − β)/2 /ω.Then, we can derive the analogue of eq.(3.14),For the illustration of the influence of diagonal magnetic moments on neutrino spin-flavor oscillations in a plane electromagnetic wave, we consider ν eL → ν µR with the same parameters as in section 3.In figure 2, we show P ν eL →ν µR based on the numerical solution of eq. ( 4.3) for different µ ′ (or ǫ 0 = µ ′ B 0 /ω) and ω = 10 13 s −1 .The upper and lower envelope functions, as well as the averaged transition probability, are also depicted in figure 2. To build the upper and lower envelope functions we use the spline interpolation of the maxima and minima of P ν eL →ν µR respectively.
Basing on figures 2(b)-2(d), one can conclude that the averaged transition probability diminishes if one accounts for µ ′ > 0 in the system.Thus the situation when µ ≫ µ a , studied in section 3, is more preferable from the point of view of phenomenological applications.
We also mention that here we study only the situation when ω = 10 13 s −1 .The numerical solution of eq. ( 4.3) in a more realistic case with ω = 10 3 s −1 , is quite problematic since eq.( 4.3) is extremely stiff then.

Conclusion
In conclusion, we mention that, in the present work, we have studied spin and spin-flavor oscillations of Dirac neutrinos in a plane electromagnetic wave with circular polarization.Our analysis was based on solving the initial condition problem for the system of flavor neutrinos interacting with external fields.This method was reviewed in ref. [13].The neutrino interaction with the electromagnetic wave is owing to the presence of the nonzero neutrino magnetic moments.This interaction was accounted for by using the exact solution of the Dirac-Pauli equation for massive fermions found in ref. [10].
In section 2, we have considered neutrino spin oscillations in a plane electromagnetic wave within one neutrino mass eigenstate.We have obtained the exact expression for the transition probability for ν L → ν R oscillations in eq.(2.5).This transition probability turns out to depend on both time and the spatial coordinate.However, for an ultrarelativistic neutrino, which is localized along its trajectory, the spatial coordinate dependence is transformed to the dependence on time only.In this case, we have reproduced the transition probability obtained in ref. [6], where the quasiclassical approach for the description of the neutrino spin evolution in electromagnetic fields was developed.
Then, in sections 3 and 4, we have generalized the approach in section 2 to treat neutrino spin-flavor oscillations.In section 3, we have started with the analysis of the situation when only a great transition magnetic moment is present in the mass eigenstates basis.In this situation, one can find the analytic expression for the transition probability; cf.eq.(3.19).Comparing the transition probability in figure 1 with the results of ref. [19], one can see that spin-flavor oscillations of Dirac neutrinos in an electromagnetic wave are analogous to oscillations in a twisting magnetic field.
The expression for the transition probability in eq.(3.19) turns out to differ from the analogous result obtained in ref. [6].This discrepancy is due to the incorrect treatment of transitions between different neutrino flavor eigenstates in ref. [6].As we have mentioned above, the neutrino spin evolution within one neutrino mass eigenstate was accounted for correctly in ref. [6].Thus, in the present work, we have provided the valid generalization of the results of ref. [6] for the description of neutrino spin-flavor oscillations in an electromagnetic wave.
Then, in section 3, we have considered an astrophysical application of the obtained results.We have studied ν eL → ν µR oscillations in an electromagnetic wave emitted by a highly magnetized millisecond pulsar.It has been found that the averaged transition probability can reach 35%; cf.figure 1(a).
Finally, in section 4, we have briefly discussed the influence of the diagonal magnetic moments on spin-flavor oscillations in an electromagnetic wave.We have assumed that diagonal magnetic moments, µ ′ ∼ 10 −14 µ B , are much smaller than the transition one, µ ∼ 10 −11 µ B .This value of µ ′ is in agreement with the upper bound on diagonal magnetic moments of Dirac neutrinos revealed in ref. [25].One can see in figure 2 that, if diagonal magnetic moments are taken into account, the averaged transition probability is less than in the case when only a transition magnetic moment is accounted for.