Ex) Article Title, Author, Keywords

Ex) Article Title, Author, Keywords

New Phys.: Sae Mulli 2022; 72: 761-768

**Published online** October 31, 2022 https://doi.org/10.3938/NPSM.72.761

Copyright © New Physics: Sae Mulli.

Ji Young Choi^{1*}, Jubin Park^{2†}

^{1}Department of Fire Safety, Seoyeong University, Gwangju 61268, Korea^{2}Department of Physics and Origin of Matter and Evolution of Galaxy (OMEG) Institute, Soongsil University, Seoul 06978, Korea

**Correspondence to:***E-mail: jychoi@seoyeong.ac.kr^{†}E-mail: honolov77@gmail.com^{1} Interestingly, WIMP particles have also been the main candidate for dark matter.

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License(http://creativecommons.org/licenses/by-nc/3.0) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

More mixing angles appear naturally when considering the spin-flip of a neutrino caused by an external magnetic field. For example, when considering neutrino oscillation (in matter) with two flavors, one mixing angle is sufficient to explain the oscillation, but if the oscillation includes both spins and flavors, six mixing angles are required. In this study, we discuss how several of mixing angles can be obtained in the spin-flavor oscillation of Majorana neutrinos and introduce a general method for this task. We also briefly discuss how to calculate the survival and transition probabilities of Majorana neutrinos using the obtained mixing angles.

**Keywords**: Mixing angles, Spin-Flavor Oscillation, Majorana Neutrino, Magnetic Field, Spin-ﬂip

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 them^{2}. 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.

The effect of matter on neutrino propagation is involved in the parameter _{α}

where _{α}_{F}_{α}_{n}_{e}

The effect on the transverse external magnetic field can be written by^{3}

where

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

where

where the _{ij}**IV. Appendix**. Here, we assume that there only exist the 2-flavor mixings _{T} or _{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

where **H** is mainly discussed in this paper, and is diagonalized by a suitable unitary matrix to find eigenvalues.

Assuming the mass ordering of _{i}**H** as follows :

where a unitary matrix

where

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 (

where six parameters are introduced as _{11}, _{22}, _{33}, and _{44}. Note that the matter effect (_{α}

where

Therefore, the two mixing angles

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 ^{4}

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

where

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

where **IV. Appendix**.

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

where

Subsequently, the survival probability can be calculated by

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,

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.

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

The general mixing matrix

where _{m}**H**), we consider this relation,

where _{1} and _{2}, 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.

Thus, the mixing angle is given by

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

Thus, the mixing angle is given by

and this is a well-known result of neutrino oscillation under external magnetic fields with two flavors in matter.

This research was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education (No. 2018R1D1A1B07051126, JBP) and (No. 2020R1I1A3066835, JYC).

^{1}Interestingly, WIMP particles have also been the main candidate for dark matter.

^{2}Various phenomenological applications and numerical analysis using this method are currently in progress [

^{3}As a result, in the case of highly relativistic neutrinos, spin precession is similar to a non-relativistic one under the transverse component of the magnetic field (see section 4 of Chapter 8 of Ref. [

^{4}Here, these trigonometric identities,

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