Estimation on the neutrino masses by using neutrino oscillation parameters
Wen-Jie Wu Xiang Zhou
























[]xiangzhou@whu.edu.cn



Hubei Nuclear Solid Physics Key Laboratory, School of Physics and Technology, Wuhan University, Wuhan 430072, China













	It has been shown that neutrino masses can be determined under the particle ansatz. In this paper, we give the general formulas of neutrino masses related to the neutrino oscillation parameters which show that there is a mass hierarchy transition. Using the best fit values measured by electron and muon neutrino oscillation experiments, the total neutrino mass is about 0.25 eV. According to the standard neutrino cosmology, the neutrino energy density is about 55 eV/cm^3 and Ω_m h^2>Ω_ν h^2∼ 0.0055. 








Estimation on the neutrino masses by using neutrino oscillation parameters
    Xiang Zhou
    December 30, 2023
==========================================================================










Nonzero neutrino masses are beyond the standard model which are confirmed by various oscillation experiments. The phase difference (PhD) in the oscillation probability derived under the so called “same energy” or “same momentum” ansatz in the traditional phenomenological theory of neutrino oscillations<cit.> is

    Δϕ_ji≃m_j^2-m_i^2/4E_νL=Δ m_ji^2/4E_νL,

where m_i and m_j are the mass eigenvalues, E_ν is the neutrino energy, L is the neutrino flight distance. It is only sensitive to the mass square difference Δ m_ji^2. However, the so called “particle” ansatz assumes that in the rest frame of a flavor neutrino ν_α all momentums of its mass eigenstates |ν_i⟩ are simultaneously zero<cit.>. Under the particle ansatz, the phase difference Δϕ_ji is mass measurable which is

    Δϕ_ji≃2m_ν_α(m_j-m_i)/4 E_ν_αL=Δ m^2_α ji/4 E_ν_αL,

where L is the distance and Δ m^2_α ji=2m_ν_α(m_j-m_i). In this paper, we give the general formulas of the neutrino masses related to the neutrino oscillation parameters, and estimate the total mass and the neutrino energy density of the Universe. 

In the case of 3-neutrino mixing U_α i is the element of the PMNS matrix U=[U_α i] with three mixing angles θ_12, θ_23 and θ_13 and Dirac phase δ. The PMNS matrix can be written as

    U_PMNS=
    	[ U_e 1 U_e 2 U_e 3; U_μ 1 U_μ 2 U_μ 3; U_τ 1 U_τ 2 U_τ 3;       ]=
    	[     1     0     0;     0  c_23  s_23;     0 -s_23  c_23;       ][      c_13         0 s_13e^-iδ;         0         1         0; -s_13e^iδ         0      c_13;           ][  c_12  s_12     0; -s_12  c_12     0;     0     0     1;       ],

where c_ij=cosθ_ij and s_ij=sinθ_ij. m_ν_α and m_i have the relationship as

    [ m_ν_e; m_ν_μ; m_ν_τ ]=
           [ |U_e 1|^2 |U_e 2|^2 |U_e 3|^2; |U_μ 1|^2 |U_μ 2|^2 |U_μ 3|^2; |U_τ 1|^2 |U_τ 2|^2 |U_τ 3|^2 ][ m_1; m_2; m_3 ].


    [ m_1; m_2; m_3 ]=
           [ U^2_e 1 U^2_μ 1 U^2_τ 1; U^2_e 2 U^2_μ 2 U^2_τ 2; U^2_e 3 U^2_μ 3 U^2_τ 3 ][ m_ν_e; m_ν_μ; m_ν_τ ],

where 

    [ |U_e 1|^2 |U_e 2|^2 |U_e 3|^2; |U_μ 1|^2 |U_μ 2|^2 |U_μ 3|^2; |U_τ 1|^2 |U_τ 2|^2 |U_τ 3|^2 ][ U^2_e 1 U^2_μ 1 U^2_τ 1; U^2_e 2 U^2_μ 2 U^2_τ 2; U^2_e 3 U^2_μ 3 U^2_τ 3 ]=
    	[ 1 0 0; 0 1 0; 0 0 1 ].

There are five independent conditions for U^2_α i which are 

    U^2_e 1+U^2_μ 1+U^2_τ 1    =     1, 
    U^2_e 2+U^2_μ 2+U^2_τ 2    =     1, 
    U^2_e 3+U^2_μ 3+U^2_τ 3    =     1, 
    U^2_e 1+U^2_e 2+U^2_e 3    =     1, 
    U^2_μ 1+U^2_μ 2+U^2_μ 3    =     1.

Therefore, there are only four independent elements in the matrix U^2=[U^2_α i]. In this paper, U^2_e 1, U^2_μ 1, U^2_e 2 and U^2_μ 2 are chosen. 

The total neutrino mass M satisfies 

    M=m_ν_e+m_ν_μ+m_ν_τ=m_1+m_2+m_3.

If the coefficients k=m_ν_e/M and ϵ=m_ν_μ/m_ν_e are introduced, the masses of flavor neutrinos can be written as

    m_ν_e    =     kM, 
    
    	m_ν_μ    =    ϵ kM, 
    
    	m_ν_τ    =     (1-k-ϵ k)M,

where 0<k<1, 0<ϵ k<1 and 0<1-k-ϵ k<1. For neutrino experiments a traditional mass square term in the PhD under the “same energy” or “same momentum” ansatz can be transformed to its corresponding mmPhD which are listed in Table <ref>. The value of ϵ=m_ν_μ/m_ν_e=|Δ m^2_μ 32|/|Δ m^2_e32| can be obtained by spectral analyses under the particle ansatz. 


Δ m^2_e21 and Δ m^2_μ 32 can be rewritten as


    Δ m^2_e21    =     2[(U^2_e2-U^2_e1)k^2                              
           +(U^2_μ 2-U^2_μ 1)ϵ k^2                         
           +(U^2_e1+U^2_μ 1-U^2_e2-U^2_μ 2)(k-k^2-ϵ k^2)]M^2,


    |Δ m^2_μ 32|     =     2|(1-U^2_e1-2U^2_e2)ϵ k^2       
           +(1-U^2_μ 1-2U^2_μ 2)ϵ^2k^2          
           +(U^2_e1+2U^2_e2+U^2_μ 1+2U^2_μ 2-2)(ϵ k-ϵ k^2-ϵ^2 k^2)|M^2.

It is easy to show that if ϵ=1, k=1/3 and then M→∞. Thus the masses of three flavor neutrinos ν_α can not be all the same. η is defined as the ratio of Δ m^2_e 21 and Δ m^2_μ 32 which is

    η≡Δ m^2_e21/|Δ m^2_μ 32|=A/|B|,

where

    A    ≡   U^2_e1+U^2_μ 1-U^2_e2-U^2_μ 2
           +(2U^2_e2-2U^2_e1+U^2_μ 2-U^2_μ 1
          +(2U^2_μ 2-2U^2_μ 1+U^2_e2-U^2_e1)ϵ)k 
    
    	B    ≡    (U^2_e1+2U^2_e2+U^2_μ 1+2U^2_μ 2-2) ϵ
           +((3-2U^2_e1-4U^2_e2-U^2_μ 1-2U^2_μ 2)ϵ
           +(3-2U^2_μ 1-4U^2_μ 2-U^2_e1-2U^2_e2)ϵ^2)k


Combining the results of mixing angles and mass square differences, k and M can be expressed as Eq. (<ref>) and Eq. (<ref>),

    k     =     -(U^2_e1+U^2_μ 1-U^2_e2-U^2_μ 2∓(U^2_e1+2U^2_e2+U^2_μ 1+2U^2_μ 2-2)ηϵ)
           /(2U^2_e2-2U^2_e1+U^2_μ 2-U^2_μ 1+(2(1±η)U^2_μ 2-(2∓η)U^2_μ 1+(1± 4η)U^2_e2
           -(1∓ 2η)U^2_e1± 3η)ϵ∓(3-2U^2_μ 1-4U^2_μ 2-U^2_e1-2U^2_e2)ηϵ^2),
    	
    
    	M     =    {2[(U^2_e2-U^2_e1)k^2+(U^2_μ 2-U^2_μ 1)ϵ k^2 
           +(U^2_e1+U^2_μ 1-U^2_e2-U^2_μ 2)(k-k^2-ϵ k^2)]/Δ m^2_e21}^-1/2,

where the upper operator in ∓ or ± is for B>0 and the lower operator for B<0. If the parameters of the PMNS matrix are known and ϵ=m_ν_μ/m_ν_e is obtained by measuring the ratio of m_μ 32 and m_e 32 from neutrino experiments, then the fraction factor k=m_ν_e/M and the total flavor neutrinos mass M can be calculated from Eq. (<ref>) and Eq. (<ref>). Consequently all  neutrino masses of flavor and mass eigenstate can be obtained. Fig. <ref> shows that there is a mass hierarchy transition with θ_12=34.7^∘, θ_13=8.43^∘, θ_23=30^∘ and δ=± 90^∘.    


The KamLAND experiment gave values of Δ m_21^2 and θ_12 by measuring long baseline reactor neutrinos. The fit values with only KamLAND data are Δ m_21^2=(7.54^+0.19_-0.18)× 10^-5 eV^2 and tan^2 θ_12=0.481^+0.092_-0.080<cit.>. The latest Daya Bay results based on 1230 live days of data gave sin^2 2θ_13=0.0841± 0.0027± 0.0019 and Δ m_32^2=(2.45± 0.06± 0.06)× 10^-3 eV^2 for normal hierarchy (NH) and Δ m_32^2=-(2.56± 0.06± 0.06)× 10^-3 eV^2 for inverted hierarchy (IH)<cit.>. Recently, NOνA made a precise measurement of ν_μ disappearance and gave Δ m_32^2=(+2.67± 0.11)× 10^-3 eV^2 for NH, and Δ m_32^2=(-2.72± 0.11)× 10^-3 eV^2 for IH<cit.>. Based on the parameters derived under the “same energy" or “same momentum" ansatz, the center values of corresponding parameters under particle ansatz are listed in Table <ref>. There are two sets of parameters due to the mass hierarchy under the “same energy" or “same momentum" ansatz which are distinguished using label (I) and (II) under particle ansatz. In addition, ϵ and η can be obtained and results are listed in table <ref>. With θ_12, θ_13, ϵ and η in Table <ref>, Fig. <ref> gives k and M of the parameter set (I) and (II)when δ runs from -180^∘ to 180^∘ and θ_23 runs from 30^∘ to 60^∘ according to Eq. (<ref>) and Eq. (<ref>). 



We define the compatibility of mass hierarchy so that the mass hierarchy is compatible when the mass hierarchy under the different ansatzs are the same, and incompatible when the mass hierarchy under the different ansatzs are different. It can be shown that the mass hierarchy of the parameter set (I) is compatible but the mass hierarchy of the parameter set (II) is incompatible. The estimation of the neutrino masses are only done for the mass hierarchy compatible parameter set (I). NOνA gave two statistically degenerate values of sin^2 θ_23 and δ both in the NH: sin^2 θ_23=0.404, δ=1.48π and sin^2 θ_23=0.623, δ=0.74π<cit.>. The latest T2K results show that sin^2 θ_23=0.532 and δ=-1.791 for NH<cit.>. Combining the knowledge of θ_23 and δ with parameter set (I), the absolute neutrino masses can be estimated and shown in Table <ref>. According to the standard neutrino cosmology, the ratio of neutrinos to photons is 3/11. Therefore, the contribution to the total energy density of the Universe from neutrinos is given by ρ_ν=m_totn_ν=m_tot(3/11)n_γ, where n_γ=412 cm^-3, m_tot=∑_ν(g_ν/2)m_ν, g_ν=4 for neutrinos with Dirac masses and g_ν=2 for Majorana neutrinos. The physical density of neutrinos in units of the critical density Ω_ν=ρ_ν/ρ_c can be written as Ω_ν h^2=m_tot/(94 eV), where h is the Hubble constant in units of 100 km·s^-1·Mpc^-1<cit.>. The neutrino energy density of the Universe with the Dirac neutrino assumption are listed in Table <ref>. The results show that Ω_m h^2>Ω_ν h^2∼0.0055.


We have estimated the neutrino masses based on the neutrino oscillation parameters<cit.>. The total neutrino mass is about 0.25 eV and the neutrino energy density of the Universe is about 55 eV/cm^3 and then Ω_m h^2>Ω_ν h^2∼0.0055. In the future the precise values with their uncertainties will be determined under the particle ansatz by the spectral analyses of neutrino oscillation experiments<cit.>.    







































































*


§ APPENDIX A: FORMULAS OF U^2_E 1, U^2_Μ 1, U^2_E 2 AND U^2_Μ 2.


    U^2_e 1    =    4 cosθ _12cos ^2θ _13cosθ _12cos 2 θ _23-cosδsinθ _12sinθ _13sin 2 θ _23/cosδsin 2 θ _12sinθ _13sin 2 θ _23 -3 cos 2 θ _13+4 cos 2 θ _12cos 2 θ _13cos 2 θ _23
    U^2_μ 1    =     (2 sin ^2θ _13cosδsin 2 θ _12sin 2 θ _23sinθ _13+2 sin ^2θ _12sin ^2θ _13cos ^2θ _23
          +2 sin ^2θ _23cos ^2θ _12-4 sin ^2θ _12cos ^4θ _13cos ^2θ _23) 
          /(cosδsin 2 θ _12sinθ _13sin 2 θ _23 -3 cos 2 θ _13+4 cos 2 θ _12cos 2 θ _13cos 2 θ _23) 
    U^2_e 2    =    -4 sinθ _12cos ^2θ _13cosδsinθ _13sin 2 θ _23cosθ _12+sinθ _12cos 2 θ _23/cosδsin 2 θ _12sinθ _13sin 2 θ _23 -3 cos 2 θ _13+4 cos 2 θ _12cos 2 θ _13cos 2 θ _23
    U^2_μ 2    =    2 sin ^2θ _13cosδsin 2 θ _12sinθ _13sin 2 θ _23-2 sin ^2θ _12sin ^2θ _23+4 cos ^2θ _12cos 2 θ _13cos ^2θ _23/cosδsin 2 θ _12sinθ _13sin 2 θ _23 -3 cos 2 θ _13+4 cos 2 θ _12cos 2 θ _13cos 2 θ _23




	X. Z. thanks the helpful discussions of Dr. Zhenyu Zhang and Dr. Qian Liu. This work is supported by the Major Program of the National Natural Science Foundation of China (Grant No. 11390381).




90
Thomson M. Thomson, Modern Particle Physics, Cambridge University Press, Cambridge (2013).
ZhouXmmPhD X. Zhou, arXiv:1802.09270 [hep-ph] (2018).
KamLAND2014A. Suzuki,

Eur.Phys.J. C74, 3094 (2014).
DayaBay2017 F. P. An et al. (Daya Bay Collaboration), Phys. Rev. D 95, 072006 (2017).
NOVA2017 P. Adamson et al. (NOνA Collaboration), Phys. Rev. Lett. 118, 151802 (2017).
NOVA2017PRL2 P. Adamson et al. (NOνA Collaboration), Phys. Rev. Lett. 118, 231801 (2017).
T2K2017 K. Abe et al. (T2K Collaboration), Phys. Rev. Lett. 118, 151801 (2017).
PDG2016 C. Partignai et al. (Particle Data Group), Chin. Phys. C, 40, 100001 (2016) and 2017 update (http://pdg.lbl.gov/2017/).
JUNO2016 F. P. An et al. (JUNO Collaboration), J.Phys. G: Nucl. Part. Phys. 43, 030401 (2016).

