6.3. SOLAR NEUTRINO PROBLEM 353
Neutrino masses
As masses are much less than kinetic energyc|p|, by the Einstein triangular relation of
energy-momentum
E^2 =p^2 c^2 +m^2 c^4 ,
we obtain an approximate relation:
E≃ |p|c+1
2
m^2 c^3
|p|.
The eigenvaluesλkof (6.3.18) andEsatisfy
(6.3.27) λk=Ek≃ |p|c+
1
2
m^2 kc^3
|p|fork= 1 , 2 , 3.Then we have
(6.3.28) λi−λj=Ei−Ej≃
(m^2 i−m^2 j)
2 Ec^4 , E≃ |p|c.By (6.3.28), if we can measure the energy differenceλi−λj, then we get the mass square
difference ofνiandνj:
∆ij=m^2 i−m^2 j.
There are three mass square differences forν 1 ,ν 2 ,ν 3 :
(6.3.29) ∆ 21 =m^22 −m^21 , ∆ 32 =m^23 −m^22 , ∆ 31 =m^23 −m^21 ,
only two of which are independent(∆ 31 =∆ 32 +∆ 21 ).
Now, we consider the mass relation betweenνe,νμ,ντandν 1 ,ν 2 ,ν 3. Applying the Dirac
operatorHˆon both sides of (6.3.19), by (6.3.18), we have
(6.3.30) Hˆ
νe
νμ
ντ
=A
λ 1 0 0
0 λ 2 0
0 0 λ 3
ν 1
ν 2
ν 3
,
whereAis the MNS matrix (6.3.26). By Postulate 5.4 (i.e. (6.2.4)), the energiesEe,Eμ,Eτof
νe,νμ,ντare given by
(6.3.31)
Ee=∫
ν∗eHˆνedx=∫
A∗ 1 kA 1 jνk∗Hˆνjdx (Hˆνj=λjνj),Eμ=∫
ν∗μHˆνμdx=∫
A∗ 2 kA 2 jνk∗Hˆνjdx,Eτ=∫
ντ∗Hˆντdx=∫
A∗ 3 kA 3 jνk∗Hˆνjdx,whereAijare the matrix elements ofA,A∗ijare the complex conjugates ofAij. The masses
me,mμ,mτofνe,νμ,ντare as follows
(6.3.32) Ee^2 =p^2 c^2 +m^2 ec^4 , E^2 μ=p^2 c^2 +m^2 μc^4 , Eτ^2 =p^2 c^2 +m^2 τc^4.