npsm 새물리 New Physics : Sae Mulli

pISSN 0374-4914 eISSN 2289-0041


Research Paper

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

Published online October 31, 2022

Copyright © New Physics: Sae Mulli.

Mixing Angles in Spin-Flavor Oscillation of Majorana Neutrinos

Ji Young Choi1*, Jubin Park2†

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

Correspondence to:*E-mail:
1 Interestingly, WIMP particles have also been the main candidate for dark matter.

Received: May 9, 2022; Accepted: September 3, 2022

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License( 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 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 Aα for the α particle (for the antineutrinos, Aα). It is given by

Aα2EVα where Vα=2GFNα=2GF(Yα/mn)ρ, and α=e,μ,....

where E represents the energy of neutrino, Vα the effective potential, GF the Fermi constant, ρ the density, Yα the number of α particles per nucleon, and mn the nucleon mass. For example, Ae 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

CT=2EμBT ,

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

tνeνμ ν¯e ν¯μ=12EM2νeνμ ν¯e ν¯μ=12EU m12000m220000m120000m22 U+ A νe 000 A νμ 0000A νe 0000A νμ + 000 CT 00 CT 00 CT 00 CT 000 νeνμ ν¯e ν¯μ=14E Σ˜ +2A νe δm C2θ   δm S2θ   0  2CT δm S2θ   Σ˜ +2A νμ +δm C2θ   2CT   00  2CT   Σ˜ 2A νe δm C2θ   δm S2θ 2CT   0  δm S2θ   Σ˜ 2A νμ +δm C2θ νeνμ ν¯e ν¯μ

where Σ˜=m22+m12, δm=m22m12, C2θ=cos2θ, S2θ=sin2θ, and the matrix U, which represents the mixing in vacuum, can be written as

U=R34(θ)R12(θ) ,

where the Rij 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 (νeνμ,  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 (CT or BT) 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

tνeνμ ν¯e ν¯μ=Σ+V νe Δm C2θ Δm S2θ 0μBT Δm S2θ Σ+V νμ +Δm C2θ μBT 00μBT ΣV νe Δm C2θ Δm S2θ μBT 0 Δm S2θ ΣV νμ +Δm C2θ νeνμ ν¯e ν¯μ=Hνeνμ ν¯e ν¯μ

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 Mi, namely M1M2 and M3M4, 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,

V=R24(θ24mb)R13(θ13mb)R23(θ23mb)R14(θ14mb)R34(θ34mb)R12(θ12mb) =V24,13,23,14V34,12 ,

where V24,13,23,14R24(θ24mb)R13(θ13mb)R23(θ23mb)R14(θ14mb) and V34,12R34(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),

ν1mbν2mb ν¯1mb ν¯2mb=V(θ12mb,θ34mb,θ14mb,θ23mb,θ13mb,θ24mb)νeνμ ν¯e ν¯μ .

2. Mixing angles

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

V24,13,23,14HV24,13,23,14=V34,12HdiagV34,12 .

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, H12HΔS, H14HμB, H22, H33, and H44. Note that the matter effect (Aα) 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

H˜(1,3)+H˜(2,4)=12C(θ14mb+θ23mb)Δ31 HS2θ13mb+Δ42 HS2θ24mb+2HμBS(θ14mb+θ23mb)S(θ13mb+θ24mb)=0 , H˜(1,3)H˜(2,4)=12C(θ14mbθ23mb)Δ31 HS2θ13mb+Δ42 HS2θ24mb+2HΔSC(θ13mbθ24mb)S(θ14mbθ23mb)=0 , 

where H˜(i,j) denotes the (i,j) component of the VHV on the left side of Eq. (10), and ΔijHHjjHii is defined similarly to the above Δ. Using the above two equations, we obtain the following two constraint equations for the mixing angles

tan(θ14mb+θ23mb) =Δ31HS2θ13mb +Δ42HS2θ24mb 4HμBS(θ13mb+θ24mb) F+(θ13mb,θ24mb) , tan(θ14mbθ23mb)=Δ31HS2θ13mb Δ42HS2θ24mb 4HΔSC(θ13mbθ24mb) F(θ13mb,θ24mb) .

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

2θ14mb=tan1F++tan1F , 2θ23mb=tan1F+tan1F ,

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

H˜(1,4)=C2θ14mb HμBC(θ13mb+θ24mb) HΔSS(θ13mbθ24mb)14S2θ14mbΔ21 H+Δ43 H+Δ31 HC2θ13mb+Δ42 HC2θ24mb=0 , H˜(2,3)=C2θ23mb HμBC(θ13mb+θ24mb)+ HΔSS(θ13mbθ24mb)+14S2θ23mbΔ21 H+Δ43 HΔ31 HC2θ13mbΔ42 HC2θ24mb=0 .

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

tan2θ14mb=4HμBC(θ13mb +θ24mb )HΔSS(θ13mb θ24mb )Δ21H+Δ43H+Δ31HC2θ13mb +Δ42HC2θ24mb G(θ13mb,θ24mb), tan2θ23mb=4HμBC(θ13mb +θ24mb )+HΔSS(θ13mb θ24mb )Δ21H+Δ43HΔ31HC2θ13mb Δ42HC2θ24mb H(θ13mb,θ24mb) .

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

F++F1F+F=G , F+F1+F+F=H .

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

H˜(1,2)= HΔSC(θ14mbθ23mb)C(θ13mbθ24mb) HμBC(θ14mb+θ23mb)S(θ13mb+θ24mb)+12Δ42 HSθ14mbCθ23mbS2θ24mb+Δ31 HCθ14mbSθ23mbS2θ13mb,=12Δ21MS2θ12mb , H˜(3,4)= HΔSC(θ14mbθ23mb)C(θ13mbθ24mb)+ HμBC(θ14mb+θ23mb)S(θ13mb+θ24mb)12Δ42 HCθ14mbSθ23mbS2θ24mb+Δ31 HSθ14mbCθ23mbS2θ13mb,=12Δ43MS2θ34mb ,

where ΔijMMiMj. 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

H˜(2,2)+H˜(1,1)=14{(Δ21H+Δ43H)(C2θ14mbC2θ23mb)2Δ31HC2θ13mb(Cθ14mb2Sθ23mb2)+2Δ42HC2θ24mb(Sθ14mb2Cθ23mb2)+4HΔSS(θ13mbθ24mb)(S2θ14mbS2θ23mb)4HμBC(θ13mb+θ24mb)(S2θ14mb+S2θ23mb)+2(Σ21H+Σ43H)}=Σ21M , H˜(4,4)+H˜(3,3)=14{(Δ21H+Δ43H)(C2θ14mbC2θ23mb)2Δ31HC2θ13mb(Sθ14mb2Cθ23mb2)+2Δ42HC2θ24mb(Cθ14mb2Sθ23mb2)4HΔSS(θ13mbθ24mb)(S2θ14mbS2θ23mb)+4HμBC(θ13mb+θ24mb)(S2θ14mb+S2θ23mb)+2(Σ21H+Σ43H)}=Σ43M , H˜(2,2)H˜(1,1)=14{(Δ21H+Δ43H)(C2θ14mb+C2θ23mb)+2Δ31HC2θ13mb(Cθ14mb2+Sθ23mb2)2Δ42HC2θ24mb(Sθ14mb2+Cθ23mb2)4HΔSS(θ13mbθ24mb)(S2θ14mb+S2θ23mb)+4HμBC(θ13mb+θ24mb)(S2θ14mbS2θ23mb)}=Δ21MC2θ12mb , H˜(4,4)H˜(3,3)=14{(Δ21H+Δ43H)(C2θ14mb+C2θ23mb)2Δ31HC2θ13mb(Sθ14mb2+Cθ23mb2)+2Δ42HC2θ24mb(Cθ14mb2+Sθ23mb2)4HΔSS(θ13mbθ24mb)(S2θ14mb+S2θ23mb)+4HμBC(θ13mb+θ24mb)(S2θ14mbS2θ23mb)}=Δ43MC2θ34mb ,

where ΣijHHii+Hjj and ΣijMMi+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

P(νeνe)=1000VeiE1t0000eiE2t0000eiE3t0000eiE4tV10002=ZP 2 ,

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

ZP(νeνe)=Z1(E1,θ12mb,θ13mb,θ14mb,θ23mb)+Z2(E2,θ12mb,θ13mb,θ14mb,θ23mb)+Z3(E3,θ13mb,θ14mb,θ23mb,θ34mb)+Z4(E4,θ13mb,θ14mb,θ23mb,θ34mb) ,


Z1=eiE1t C θ 12 mb C θ 13 mb C θ 14 mb +S θ 12 mb S θ 13 mb S θ 23 mb 2 ,Z3=eiE3t C θ 23 mb C θ 34 mb S θ 13 mb C θ 13 mb S θ 14 mb S θ 34 mb 2 , Z2=eiE2t C θ 13 mb C θ 14 mb S θ 12 mb C θ 12 mb S θ 13 mb S θ 23 mb 2 ,Z4=eiE4t C θ 13 mb C θ 34 mb S θ 14 mb +C θ 23 mb S θ 13 mb S θ 34 mb 2 .

Subsequently, the survival probability can be calculated by

P(νeνe)=|Z1|2+|Z2|2+|Z3|2+|Z4|2+2Z1Z2*+Z1Z3*+Z1Z4*+Z2Z3*+Z2Z4*+Z3Z4* .

As [Z1Z2*] [ei(E2E1)t] cos(E2E1)t, all interference terms have an energy difference. Thus define a wavelength for the oscillation as

λijmb=2π/|MiMj|=2π/|ΔijM| ,

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

E2E1=(m22+p2)1/2(m12+p2)1/2(m22m12)/2p2π/λ .

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 θ12mbθ12mθm, and ZP is simply given by

ZP=eiE1tCθm2+eiE2tSθm2 .

The survival probability in this case is

P(νeνe)=112sin22θm1cos(E2E1)t ,

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.

R(θ12)=cosθ12 sinθ12 00sinθ12 cosθ12 0000000000,R(θ34)=0000000000cosθ34 sinθ34 00sinθ34 cosθ34 ,R(θ23)=00000cosθ23 sinθ23 00sinθ23 cosθ23 00000 , R(θ14)=cosθ14 00sinθ14 00000000sinθ14 00cosθ14 ,R(θ13)=cosθ13 0sinθ14 00000sinθ13 0cosθ13 00000,R(θ24)=0cosθ24 0sinθ24 00000sinθ24 0cosθ24 0000 .

2. Cases of mixing with 2 flavors

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

U=cosθmsinθmsinθmcosθm ,

where θm 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,

H=UHDU=cosθmsinθmsinθmcosθmM100M2cosθmsinθmsinθmcosθm=M1Cθm2+M2Sθm212(M2M1)S2θm12(M2M1)S2θmM1Sθm2+M2Cθm2 ,

where C2θmcos2θm, S2θmsin2θm, Cθm2cos2θm and Sθm2sin2θm. The two eigenvalues M1 and M2, and their difference are given by

M1=12H11+H22(H11 H22 )2+4H12H21,  M2=12H11+H22+(H11 H22 )2+4H12H21, Δ21MM2M1=(H11 H22 )2+4H12H21 .

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

H11+H22=M1+M2 H11H22=(M1M2)(Cθm2Sθm2)=Δ21MC2θm H12=H21=12(M2M1)S2θm=12Δ21MS2θm .

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 :

S2θm=2H12Δ21M=2H21Δ21M , C2θm=H22H11Δ21M .

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 νeνμ in matter}

H2x2=Σ+VνeΔmC2θΔmS2θΔmS2θΣ+Vνμ+ΔmC2θ . Δ21M=(VνeVνμ)2ΔmC2θ2+4Δm2S2θ2

Thus, the mixing angle is given by

S2θm=ΔmS2θ(Vνe Vνμ )/2Δm C2θ 2+Δm2S2θ2 ,

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)

H2x2=Σ+VνeΔmμBTμBTΣ+Vνμ+Δm . Δ21M=(VνeVνμ)2Δm2+4μ2BT2

Thus, the mixing angle is given by

S2θmb=μBT(Vνe Vνμ )/2Δm 2+μ2BT2 ,

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).

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

2Various phenomenological applications and numerical analysis using this method are currently in progress [25]. Especially in celestial bodies with extreme conditions such as supernovae, neutron stars, and magnetas, this effect can be very significant and can lead to remarkable results.

3As 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. [24] for details).

4Here, these trigonometric identities, tan(α±β)=(tanα±tanβ)/(1tanαtanβ) are used.

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