6.3. SOLAR NEUTRINO PROBLEM 357
whereνk( 1 ≤k≤ 3 )are the two-component Weyl spinors, and~σ= (σ 1 ,σ 2 ,σ 3 )
Based on the massless model (6.3.44), both problems of parity and handedness of neutri-
nos have been resolved, and we can derive in the same conclusions as given in (6.3.24) and
(6.3.25). In this case, the differencesλi−λjof eigenvalues in the transition probabilities such
as (6.3.24) and (6.3.25) stand for the differences of frequencies:
(6.3.45) λi−λj=ωi−ωj,
whereωk( 1 ≤k≤ 3 )are the frequencies ofνk.
However, the massless model also faces the problem of infinite number of eigenvalues
as mentioned above. The eigenvalue equations in (6.3.44) for the straight line motion on the
y−axis is written as
(6.3.46) ihc ̄α^2
d
dy(
φ^1
φ^2)
=λ(
φ^1
φ^2)
with α^2 =(
0 −i
i 0)
.
The eigenvalues of (6.3.46) are
(6.3.47) λk=khc ̄ , ∀k> 0 ,
and each eigenvalue of (6.3.47) has two eigenstates
(6.3.48)
(
φ 11
φ 12)
=
(
sinky
−cosky)
,
(
φ 21
φ 22)
=
(
cosky
sinky)
.
The eigenvalues of (6.3.44) atx-axis andz-axis are all the same as in (6.3.47), and the
eigenstates at thexandzaxes are
(6.3.49)
(
φ^1
φ^2)
=
(
e−ikx
e−ikx)
and(
φ^1
φ^2)
=
(
e−ikz
0)
or(
0
eikz)
.
6.3.6 Neutrino non-oscillation mechanism
Although the massless neutrino oscillation model can solvethe parity and the handedness
problems appearing in the massive neutrino oscillation mechanism, the problem of infinite
numbers of eigenvalues and eigenstates still exists in the model (6.3.44).
In fact, the weakton model first introduced in (Ma and Wang,2015b) can provide an
alternative explanation to the solar neutrino problem. Based on the weakton model, there
exists aν-mediator, whose weakton constituents are given by
(6.3.50) ν=αeνeνe+αμνμνμ+ατντντ,
whereαe^2 +αμ^2 +αμ^2 =1. The valuesαe^2 ,α^2 μ,ατ^2 represent the ratio of the neutrinosνe,νμ
andντin our Universe
In view of (6.3.50), we see the reaction
(6.3.51)
νe+νe−→ν (νeνe),
νμ+νμ−→ν (νμνμ),
ντ+ντ−→ν (ντντ),