The first results of the solar neutrino experiment were presented in 1968[1], which conflicted with the accepted star evolution model. According to reports at the time, the observed neutrino rate[2] was approximately one-third to one-half of the theoretical prediction[3]. The discrepancy between these experimental results and the predictions of the standard solar model[4-6] has persisted to this day, and is now well known as the solar neutrino problem or puzzle.

Several possible solutions have been proposed so far to solve this interesting problem or puzzle. The first class of solutions involves neutrino oscillations[7-11] or neutrino magnetic moment[12-14], or a combination of both[15-17]. The second class, on the other hand, deals with the hypothetical neutral heavy stable particles[18-20], also known as WIMPS (weakly interacting massive particles)¹. In this study, among the proposed solutions, we focus mainly on the aspects of the magnetic moments of Majorana neutrinos. Generally, as the magnetic moment of neutrinos is minute, it is difficult to expect a noticeable spin precession by the solar magnetic field[12-14].

One of the interesting features revealed in the data was the time variation in neutrino fluxes, suggesting an anticorrelation with sunspot activity. This is an approximately cyclic phenomenon with a period of approximately 22 years and is associated with regular variations in the solar magnetic field in the convection region, the outer layer. Because the sun does not rotate as a rigid body, it rotates faster near the equator, resulting in the deformation of the lines of force and a polarity reversal after reaching a maximum. Since then, these reversals have occured periodically, with solar activity being the most active in the strongest magnetic field. At this most active time, the size and number of sunspots are maximized, and the neutrino flux is minimized. This fact triggered the revival of the magnetic moment hypothesis by Voloshin, Vysotskij, and Okun (VVO)[12-14]. It was emphasized that neutrino oscillations[7-11] can not explain the anticorrelation effect[21]; hence, the attractiveness of the VVO proposal. An interesting development of this proposal was the suggestion in 1987 by Lim and Marciano[22] and Akhmedov[23] that the neutrino spin-flip could occur anywhere inside the Sun through a resonant process. Therefore, we intend to consider well-known neutrino oscillations with the magnetic dipoles and transition moments of neutrinos. In particular, we attemp to find solutions that describe this oscillation without using previously used approximation as much as possible. As a starting point, this study focuses on how to obtain the mixing angles of the matrix describing this oscillation phenomenon. In this process, we introduce one general method for obtaining them². Before we conclude, we briefly discuss how to calculate survival and transition probabilities using the obtained mixing angles.

This paper is organized as follows. In Sec. II, we first introduce the propagation equations of neutrinos with two flavors under an external magnetic field. A Hamiltonian involving the conversion of left-handed and right-handed neutrinos is diagonalized via a unitary matrix consisting of six rotation matrices. We then demonstrate how the six angles of the rotation matrices are expressed by Hamiltonian components. Finally we show how to estimate the survival and transition probabilities of Majorana neutrinos in this case. In Sec. III we conclude.

We first introduce the evolution equation for propagating Majorana neutrinos assuming that there are only two neutrino flavors and non-negligible spin-flip effects. In particular, when the left-right handedness conversion by the external magnetic field and the magnetic moment of the Majorana neutrino is considered, the Hamiltonian that describes this process becomes twice as large as its original size. Therefore, in the case of neutrino oscillation with two flavors, it is described by the Hamiltonian extended from the previous 2-by-2 to the 4-by-4 size. Unlike the previous 2-by-2 case (with only one mixing angle), a unitary matrix with six rotation angles is required in the diagonalization process. However, this Hamiltonian has a symmetrical form with zero components due to the Majorana property, so six mixing angles can be easily expressed with the given Hamiltonian components. In this section, focusing on this point, we discuss how to express the mixing angles.

1. Evolution equation of Majorana neutrinos

The effect of matter on neutrino propagation is involved in the parameter A_α for the α particle (for the antineutrinos, −Aα). It is given by

where E represents the energy of neutrino, V_α the effective potential, G_F the Fermi constant, ρ the density, Y_α the number of α particles per nucleon, and m_n the nucleon mass. For example, A_e denotes the induced mass squared of the electron neutrino that arises from propagation via a background of electrons.

The effect on the transverse external magnetic field can be written by³

where μ describes the transition moment between different flavors. Note that Majorana particles can not have magnetic dipole moments, i.e., the transition moments between the identical flavors.

The equation for propagating Majorana neutrinos under the influence of an external magnetic field in matter becomes

where Σ˜=m22+m12, δm=m22−m12, C2θ=cos2θ, S2θ=sin2θ, and the matrix U, which represents the mixing in vacuum, can be written as

where the R_ij represents a rotation matrix on the i-j plane, and for a detailed form, see subsection 1 in IV. Appendix. Here, we assume that there only exist the 2-flavor mixings (νe↔νμ,  ν¯e↔  ν¯μ) in a vacuum. Note that mνe(μ)=m ν¯e(μ). Of course, it is easy to see that within the limit where the magnetic field (C_T or B_T) disappears, this 4-by-4 matrix is reduced to two well-known 2-by-2 forms. It is also interesting that due to the transition magnetic moments of Majorana neutrinos, the non-zero external magnetic field can mix νe(νμ) and ν¯μ(ν¯e). They correspond to R14(R23) rotation. To simplify, divide the above 4-by-4 matrix by the energy of 4E to obtain the following form

where Δm=δm/4E and Σ=Σ˜/4E. This form of Hamiltonian H is mainly discussed in this paper, and is diagonalized by a suitable unitary matrix to find eigenvalues.

Assuming the mass ordering of M_i, namely M1≤M2 and M3≤M4, we can diagonalize the matrix H as follows :

where a unitary matrix V is introduced and it can be parameterized by the product of six rotation matrices as below,

where V24,13,23,14≡R24(θ24mb)R13(θ13mb)R23(θ23mb)R14(θ14mb) and V34,12≡R34(θ34mb)R12(θ12mb). Note that in this parameterization, for simplicity, we have neglected the possible presence of CP-violating phases. The matrix V also associates flavor eigen states (νe,μ,  ν¯e,μ) with mass-magnetic eigen states (ν1,2mb,  ν¯1,2mb),

2. Mixing angles

To express the mixing angles as components of a given Hamiltonian H, we consider the following relation

Because the matrix components of the left and right expressions must be equal, we can derive the relations for the six mixing angles (θ12mb,θ34mb,θ14mb,θ23mb,θ13mb,θ24mb). From Eq. (5), we can assign the typical Hamiltonian for the Majorana neutrinos as follows;

where six parameters are introduced as H₁₁, H12≡HΔS, H14≡HμB, H₂₂, H₃₃, and H₄₄. Note that the matter effect (A_α) is related to the diagonal components of the Hamiltonian matrix, and the effect (μBT) by magnetic moments is related to the non-diagonal components. From Eq. (10), we first consider the following relations

where H˜(i,j) denotes the (i,j) component of the V†HV on the left side of Eq. (10), and ΔijH≡Hjj−Hii is defined similarly to the above Δ. Using the above two equations, we obtain the following two constraint equations for the mixing angles

Therefore, the two mixing angles (θ14mb,θ23mb) can be expressed in terms of the angles (θ13mb,θ24mb)

From the (1,4) and (1,3) components, we can also consider the following two relations

After some arrangement of terms, we can derive these two additional relations for the mixing angles

Using Eq. (16), (20), and (17), and (21), we obtain only the useful equations for the mixing angles (θ13mb,θ24mb)⁴

The (1,2) and (3,4) components can be written by

where ΔijM≡Mi−Mj. After fixing the above four mixing angles (θ13mb,θ24mb,θ14mb,θ23mb) through the above relations, we can derive the relations for the two remaining mixing angles, θ12mb and θ34mb. They correspond to the typical mixing angles θ12m and θ34m, respectively, in matter when the magnetic effect disappears (μBT=0).

Finally, important relations for the mixing angles are derived using the Hamiltonian diagonal components

where ΣijH≡Hii+Hjj and ΣijM≡Mi+Mj. Thus, given the special form of the Hamiltonian for Majorana neutrinos, we can obtain the required mixing angles using the above-mentioned relations. For the 2-by-2 matrix obtained under the special limits of previous studies, see subsection 2 in IV. Appendix.

3. Survival and transition probabilities

In this subsection, we only consider the survival probability of electron Majorana neutrino. Other survival and transition probabilities among the remaining neutrinos can be estimated in the same way as discussed here. The survival probability is given by

where V represents the 4-by-4 unitary matrix containing 6 rotation angles we obtained in the previous subsection, (θ12mb,θ34mb,θ14mb,θ23mb,θ13mb,θ24mb), and ZP is a complex quantity to describe the survival probability. Generally, ZP can be written by

where

Subsequently, the survival probability can be calculated by

As ℜ[Z1Z2*] ℜ[ei(E2−E1)t] cos(E2−E1)t, all interference terms have an energy difference. Thus define a wavelength for the oscillation as

thus, each interference term has its own wavelength. Recall that in the case of vacuum oscillation, the energy difference and the wavelength can be given by

In the case of neutrino oscillation without the external magnetic field, θ13mb=θ14mb=θ23mb=θ24mb=θ34mb=0, and the associated non-zero mixing angle becomes θ12mb→θ12m≡θm, and ZP is simply given by

The survival probability in this case is

and this result is consistent with the well-known result in oscillation (in matter) with 2 flavors.

We consider the well-known neutrino oscillations under an external magnetic field. Here, Majorana neutrino contains the transition moments, and its spin-flip occurs. More mixing angles appear naturally caused by the increased Hamiltonian matrix. The diagonalization of the given unitary matrix Hamiltonian is also more complicated form than before. In this paper, we particularly focus on the various mixing angles of the unitary matrix, and discuss how to obtain these mixing angles. Finally, survival and transition probabilities are briefly discussed on the basis of these mixing angles.

1. Four-dimensional rotation matrices

The six rotation matrices discussed in the main text are defined as follows.

2. Cases of mixing with 2 flavors

The general mixing matrix U in 2 neutrino flavors is given by

where θ_m denotes a mixing angle in matter. Note that this may differ from the mixing angle(θ) in a vacuum. To express the mixing angle relation in terms of components of the given Hamiltonian(H), we consider this relation,

where C2θm≡cos2θm, S2θm≡sin2θm, Cθm2≡cos2θm and Sθm2≡sin2θm. The two eigenvalues M₁ and M₂, and their difference are given by

From the above Eq. (39), we can obtain the following useful relations

The first equation simply shows the conservation of the trace under the unitarity transformation, and the second and third equations express the desired relations for the mixing angle. Using these, we obtain the sine and cosine values of the mixing angle :

As simple examples, we consider the following two cases : The first case represents the Hamiltonian of the usual neutrino oscillation in matter, and the second case represents the Hamiltonian at a limit, where the vacuum-mixing angle is zero.

1) Mixing in νe−νμ in matter}

Thus, the mixing angle is given by

and this is a well-known result of a neutrino oscillation with 2 flavors in matter.

2) Mixing in νe− ν¯μ under an external magnetic field with a vanishing vacuum mixing (θ = 0)